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%            Adjoint vector estimates and subanalyticity of 
%          the distance function for real-analytic strongly
%             bracket-generating subriemannian metrics
%
%            by Hector J. Sussmann and Andre Bellaiche
%
%             Date of this version:August 7, 1998
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%    This version is still a DRAFT:  DO NOT QUOTE WITHOUT PERMISSION
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%            THIS VERSION SUPERSEDES ALL PREVIOUS VERSIONS 
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%            The text is 50 pages long, plus a title page
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{\mymagonerm

\vskip1cm 

\noindent Andr\'e: 

\bigbreak


\noindent Here is the product of my latest efforts on the paper.
There still  are some missing things:

\bigbreak

\item{1.} The estimate of Proposition 3.12 is crucial for the proof of our
main theorem. I think that in the references I give there ---to work by
Gershkovich and Vershik--- the estimate is proved only for the case of a
nonsingular distribution. (I confess that I have {\mymagoneit not} looked
at the Gershkovich-Vershik papers.)  Of course, the estimate is also valid
for singular distributions, and the proof is fairly trivial ---though a
little bit long--- and I am sure there are references where it is done.
Could you include such a reference?  (For example, it's probably done in
one of your papers. If so, then we should put that in.) 

\item{2.} Add references on subanalyticity of the distance function 
in  the Riemannian case, if you can find any. 


\item{3.} Add mention of stuff by Agrachev and Sarychev on subanalyticity 
for the 2-generating non SBG case.

\item{4.}  Also, I guess there is work by Sebastien Jacquet. His e-mail
address is

\medbreak\centerline{Sebastien.Jacquet@univ-savoie.fr}

\medbreak
\item{} You should try to get a copy and include a mention of it.  

\item{5.} Include an example of a nonsubanalytic distance function for a
  nonconvex Riemannian manifold (the plane minus a sequence of
  holes) if you think this can be done without making it too long. 

\item{6.} Give reference (e.g. Dixmier) on metric spaces
where $d$ can have infinite values. 

\item{7.} Put your complete address in the front page.

\item{8.} If you want to put a footnote with some acknowledgement, 
do it. 

\item{9.} Add other bibliographical references that you might think are
needed, for example to your own papers. 

\bigbreak\noindent In addition, you should {\mymagonebf proofread the whole
paper several times}, to make sure we get rid of mathematical mistakes and
typographical errors. 

\bigbreak\noindent Do you think you could finish the work, say, within one
month from today, by {\mymagtwobf August 17}  or so?  

\bigbreak\noindent Also, would you be willing to take care of all the work
from now on (sending the paper to the journal, dealing with the editors,
etc.)?

\bigbreak\noindent Cordially, 

\bigbreak\noindent ---Hector 
}

\vfill\eject

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\pltwocen{sussmann@hamilton.rutgers.edu}{abellaic@math.jussieu.fr}

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\hectorandre
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\noindent{\mysmall ${\,}^{*}$\ \ Supported in part by 
NSF Grant DMS95-00798 and AFOSR Grant 0923.  The final part of the
work was carried out between January and June 1998, while this author
was a visitor at the Institut Henri Poincar\'e (IHP), in Paris. He is
grateful to the IHP for its kind hospitality during this period.
}

}
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\def\myrunninghead{Adjoint vector estimates}
\def\theshortyear{92}
\def\thelongyear{1998}
\def\themonth{July}
\def\thesyconreportnumber{03}
\def\thetitleone{ADJOINT VECTOR ESTIMATES, CONVEXITY,} 
\def\thetitletwo{AND SUBANALYTICITY OF THE DISTANCE FUNCTION}
\def\thetitlethree{FOR STRONGLY BRACKET-GENERATING} 
\def\thetitlefour{SUB-RIEMANNIAN METRICS}

\def\myabstract{
\noindent We prove  that, for a smooth strongly bracket-generating 
subriemannian structure on a smooth manifold $M$, the set
${\cal G}_{nc,min}(K)$  of all curves
$\Xi:[0,1]\to T^*M$ in the cotangent bundle $T^*M$ that are solutions of
the ``geodesic equation'' and project down to a nonconstant length
minimizer contained in a given compact set $K$ is itself relatively
compact.  As a consequence, we show that, if $\bar K$ and $\hat K$ are
compact subsets of $M$, and $U$ is a neighborhood of $\bar K$ such that for
every pair $(x,y)\in U\times\hat K$ there exists a length minimizer joining
$x$ to $y$, then there exists a compact set ${\cal K}$ of solutions
$\,\,\Xi:[0,1]\to T^*M$ of the geodesic equation such that every length
minimizer $\xi:[0,1]\to M$ for which $\partial\xi=(\xi(0),\xi(1))\in\bar
K\times\hat K$ is the projection of a $\Xi\in{\cal K}$. In the
real-analytic case, this implies that the distance function $d:M\times
M\to\rr$ is subanalytic if $M$ is convex (in the sense that any two points
of $M$ can be joined by a length minimizer) and, more generally, that the
restriction $d\restr\Omega$ of $d$ to $\Omega$ is subanalytic in $\Omega$
---that is, the graph of $d\restr\Omega$ is a subanalytic subset of
$\Omega\times\rr$--- if $\Omega$ is an open subset of $M\times M$ such that
for every $(\bar x,\hat x)\in\Omega$ there exists a length minimizer from
$\bar x$ to $\hat x$. In particular, $d$ is subanalytic on a neighborhood
of the diagonal, and $d$ is subanalytic on $U\times U$ if $U$ is any open
subset of $M$ such that any two points of $U$ can be joined in $M$ by a
length minimizer.  

}


\xdef\listref{\line{\noindent {\bf REFERENCES}\hfill}
\bigbreak}
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\medbreak
\vbox{
\nitembfnoadv A.~A. Agrachev and A.~V.~Sarychev, ``Abnormal sub-Riemannian
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\nitembfnoadv A.~A. Agrachev, B.~Bonnard, M.~Chyba and I.~Kupka,
``Sub-Riemannian sphere in Martinet flat case,'' 
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\xdef\vershgeb{\the\myitemno}


\headline={\myownsyconheadline}
\xdef\theabstract{\myabstract}

\notetoandre

\pageno=0

\null
\vfill\eject

\pageno=0
\thestart

\vfill\eject

\pageno=0
\null

\vfill\eject

\section{Introduction}

\noindent
The concept of a ``strongly bracket-generating'' (SBG) subriemannian
structure was introduced by  R. Strichartz in [\stricha] and [\strichb],  
in order to single out a natural class of subriemannian manifolds for which 
nonconstant abnormal minimizers do not occur. 

The main purpose of this paper is to prove the following estimate on the
fields of covectors corresponding to energy-minimizing arcs in a strongly
bracket-generating subriemannian manifold ${\cal M}$: if $K$ is a compact
set of points of ${\cal M}$, and ${\cal G}_{nc,min}(K)$ is the set of all
solutions $\Xi$ of the geodesic equation for ${\cal M}$ that are defined on
$[0,1]$ and project down to a nonconstant energy minimizer contained in
$K$, then ${\cal G}_{nc,min}(K)$ is relatively compact. This has a number
of important consequences, one of which is the fact that, if an open set
$U$ is ``relatively convex'' (that is, any two points of $U$ can be joined
by a minimizer, which need not be contained in $U$) then for every compact
subset $K$ of $U$ the set of solutions of the geodesic equation that are
defined on $[0,1]$ and project down to a nonconstant energy minimizer whose
endpoints are in $K$ is also relatively compact.  This yields as a simple
corollary the subanalyticity of the distance function for real-analytic
subriemannian manifolds that satisfy a geodesic convexity condition.  
(Global subanalyticity of the distance to a
point $\bar x$, for the 3-dimensional contact case, 
on any punctured ball $\{x:0<d(x,\bar x)\le R\}$ such that 
the closed ball $\{x:0\le d(x,\bar x)\le R\}$ is compact, 
was proved in Sussmann [\sussioffe]. A local result in the general 
SBG case, giving subanalyticity on some small neighborhood of a point,
can be found in Ge [\zge].  More recently, an example of 
a non-SBG real-analytic subriemannian manifold for which there are 
points $x$ such that the $r$-ball centered at $x$ fails to be 
subanalytic even for small $r$ has been exhibited by 
Agrachev, Bonnard, Chyba and Kupka in [\chyba].)

As in Bella\"\i{}che [\bellaic], we work in a slightly more general setting
than that usually considered in 
the literature (cf. for example Agrachev-Sarychev [\agrachev],
B\"ar [\cbar], Brockett
[\brockaa,\brocka], Bryant-Hsu [\bryant], Gaveau [\gaveau], Ge [\zge],
Gershkovich [\gershko,\gershkobb], Hamenst\"adt [\hamen], Kupka [\kupka], 
Liu-Sussmann [\liusslee,\liussmem], Montgomery [\montgo,\montgob], Strichartz
[\stricha,\strichb], Sussmann [\cornu], Taylor [\taylora,\taylorb],
Vershik-Gershkovich [\vershgea,\vershgeb]). 
Instead of defining a ``subriemannian manifold''
to be a triple $(M,E,G)$ such that $M$ is a manifold, $E$ is a subbundle of
the tangent bundle $TM$, and $G$ is a Riemannian metric on $E$, we allow
$E$ to be a ``singular distribution,'' i.~e. a pair $(F,\sigma)$ such that
$F$ is a smooth vector bundle over $M$ and $\sigma=\{\sigma_x\}_{x\in M}$
is a smooth vector-bundle homomorphism from $F$ to $TM$, without 
requiring all the maps $\sigma_x$ to have the same rank.  On the other
hand, we follow the customary practice of incorporating the
``bracket-generating'' property ---also known as the ``H\"ormander
condition''--- into the definition. 

Exactly as in the Riemannian case, the length minimization problem in a
subriemannian manifold ${\cal M}=(M,F,\sigma,G)$ is closely related to that
of energy minimization for arcs $\xi:[0,1]\to M$. Precisely, $\xi$ is an
energy minimizer if and only if it is a length minimizer parametrized by
constant times arc-length.  Since every admissible arc $\xi:[a,b]\to M$
admits a ``canonical reparametrization'' ---i.~e. a reparametrization
$\xi_1:[0,1]\to M$ which is parametrized by constant times arc-length---
it follows that one can find all the length minimizers  by finding instead 
all the energy minimizers. 

The Maximum Principle of optimal control theory (cf. Berkovitz [\berko],
Lee-Markus [\leemark], Pontryagin {\it et al.} [\pontr], and also Sussmann
[\susscofree] for a coordinate-free version particularly adapted to the
subriemannian minimization problem) implies that every energy minimizer is
an ``extremal.''  Extremals are of one of two (not mutually exclusive)
kinds, namely, ``geodesics'' (also called ``normal extremals'') or
``abnormal extremals.'' Geodesics are projections of curves in the
cotangent bundle $T^*M$ that are ``solutions of the geodesic equation,''
i.e. integral curves of a Hamilton vector field $\vec{H}_{{\cal{}M}}$ on
$T^*M$ arising from a function $H_{{\cal M}}$ ---the ``Hamiltonian'' of
${\cal M}$--- which is smooth if ${\cal M}$ is smooth, and analytic if
${\cal M}$ is analytic.  In general, it is possible for a length minimizer
to be ``strictly abnormal'' (i.~e. not a normal extremal, which implies in
particular that it is an abnormal extremal). This was a controversial issue
for more than ten years ---during which a number of erroneous proofs were
given of the nonexistence of strictly abnormal minimizers, as well as
incorrect examples of such minimizers--- but the question is now definitively
settled.  Examples of strictly abnormal minimizers are given in Montgomery
[\montgo,\montgob], Liu-Sussmann [\liusslee], [\liussmem], Sussmann [\cornu],
and Agrachev-Sarychev [\agrachev]. 
(For earlier discussions of this issue, cf.  B\"ar [\cbar], Brockett
[\brockaa,\brocka], Gaveau [\gaveau], Hamenst\"adt [\hamen], Strichartz
[\stricha,\strichb], Taylor [\taylora,\taylorb].)

If ${\cal M}$ is SBG, then nonconstant abnormal extremals do not occur.  So
the necessary condition for an arc $t\to\xi(t)$ in $M$ to be a minimizer is
that there exist a field $t\to\psi(t)$ of covectors along $\xi$ ---known in
control theory as an ``adjoint vector''--- such that the curve
$t\to\Xi(t)=(\xi(t),\psi(t))$ is a solution of the geodesic equation.  Our
main result is really an {\it a priori} bound for the adjoint vector $\psi$:
if $K\subseteq M$ is compact, then there exists a compact subset $J_K$ of
$T^*M$ such that, if $\xi:[0,1]\to M$ is any nonconstant energy minimizer
contained in $K$, parametrized by constant times arc-length, then the 
solution $\Xi=(\xi,\psi)$ of the geodesic equation that projects down to
$\xi$ (which happens to be unique, cf. Theorem 6.7 below) 
is in fact contained in $J_K$.

Our estimate is trivial for Riemannian metrics, because in that case
$||\psi(t)||$ is just the length of $\xi$. In the subriemannian case,
one automatically gets a bound in terms of the length for the
components of $\psi(t)$ in the direction of the vectors in
$E(\xi(t))$, but this is not enough, in principle, to yield an estimate
for the magnitude of the whole covector $\psi(t)$.  To get such an
estimate, one needs a more delicate argument, based on a detailed 
analysis of the geodesics and integration by parts.

The paper is organized as follows: in \S 2 we review some basic definitions
and introduce notations about manifolds, charts, curves, momentum functions
and Hamiltonian vector fields; we also define
the notion of a singular distribution, and review some of the basic
properties of these objects and of the natural classes of curves
associated to them. In \S 3 we review the definitions of subriemannian
manifold, the distance function, and length and energy minimizers.  In \S 4
we discuss the basic convexity theorems for subriemannian manifolds, and in
particular the subriemannian version of the Hopf-Rinow theorem.  In \S 5 we
review the theory of extremals and geodesics, and explain the basic facts
relating these curves to minimizers, namely, that every energy minimizer is
an extremal and every geodesic is a local minimizer.  In \S 6 we review the
definition of an SBG structure, and study some of its elementary
properties, such as (a)~the nonexistence of nonconstant abnormal extremals,
(b)~the uniqueness of the adjoint vector, (c)~the fact that the local
minimizers are exactly the geodesics, and (d)~the particularly
simpler form that the Hopf-Rinow theorem has in the SBG case. In \S 7 we
state and prove our adjoint vector estimate. Finally, in \S 8 
we use the results of \S 7 to prove theorems on subanalyticity 
of the subriemannian distance in the SBG case.   A number of 
proofs ---especially those dealing with relatively less interesting 
technical details such as measurability of various maps, 
and the precise properties of reparametrization by arc-length--- 
are relegated to an appendix. 

\section{Manifolds, singular distributions, charts, curves}

\noindent We begin by reviewing some basic concepts and well known facts,
and introducing some notations that will be used throughout the paper.

The word {\it interval} will always mean ``connected subset 
of $\rr$.''   An interval $I$ is {\it nontrivial} if it contains at least two
points, i.~e. if it has positive measure. In that case, every nonempty 
relatively open subinterval of $I$ is also nontrivial. 

We define a {\it metric space} to
be a pair $(M,d)$ where $M$ is a set and $d:M\times M\to[0,+\infty]$ is an
extended-real-valued function satisfying the usual properties ---symmetry,
the triangle inequality, and $d(x,y)\!\!=\!\!0\Longleftrightarrow
x\!\!=\!\!y\,$--- of a 
distance.  This only differs from the most widely used definition in that
we allow $d$ to take the value $+\infty$. If $(M,d)$ is a metric space in
our sense, then we can define $\tilde d(x,y)=\min\Big(d(x,y),1\Big)$, and
$(M,\tilde d)$ is another metric space having the same topology and uniform
structure as $(M,d)$. So, for example, the notion of metrizable space in
our sense agrees with the usual one.  In particular,  metrizable spaces
are always paracompact (cf. Kelley [\kelley]). 

If $M$ is any metric space, and $I$ is a compact subinterval of $\rr$, we
use $C^0(I,M)$ to denote the space of all continuous maps from $I$ to $M$,
with the topology of uniform convergence.  A sequence $\{\xi_j\}$ in
$C^0(I,M)$ converges in $C^0(I,M)$ ---i.~e. uniformly--- if and only if
$\xi_j(t_j)\to\xi(t)$ whenever $t_j\to t$.  In particular, the topology of
$C^0(I,M)$ depends only on the topology of $M$, and not on the metric,
and is therefore well defined if $M$ is just a metrizable topological space. 

The word ``smooth'' will mean ``of class $C^{\infty}$.''  The word
``manifold'' will always stand for ``smooth finite-dimensional Hausdorff
paracompact manifold without boundary.''  (In any case, we will only be
interested in subriemannian manifolds, which are always metrizable, and
hence paracompact.)

If $M$ is a manifold, we use $C^\infty(M)$ to denote the ring of smooth
real-valued functions on $M$. A {\it vector bundle} is a smooth,
finite-dimensional vector bundle.  If $F$ is a vector bundle over a
manifold $M$, and $x\in M$, we use $F(x)$ to denote the fiber of $F$ at
$x$. The {\it dimension} ${\rm dim}\,F$ of $F$ is its fiber dimension.  We
use $F^*$ to denote the dual of $F$, and $\Gamma(F)$ to denote the space of
smooth sections of $F$, so $\Gamma(F)$ is a $C^\infty(M)$-module.  We write
$\pi_{_F}$ to denote the canonical projection from $F$ to $M$.  We use
$\mylangle\cdot,\cdot\myrangle$ ---omitting the explicit mention of $F$--- to
indicate the canonical pairing of $F^*$ and $F$, so $\mylangle v,w\myrangle$ is
defined iff $v\in F^*$, $w\in F$, and $\pi_{_{F^*}}(v)=\pi_{_F}(w)$.  If
$U$ is an open subset of $M$, then $F\restr U$ denotes the restriction of
$F$ to $U$.  A {\it local basis of sections} of $F$ is a basis ${\bf
f}=(f_1,\ldots,f_m)$ of smooth sections of $F\restr U$ for some nonempty
open subset
$U$ of $M$.  If ${\bf f}$ is a local basis of sections of $F$, we write
$U_{\bf f}$ to denote the domain of ${\bf f}$.  If $x\in M$, then a {\it
local basis of sections of $F$ near $x$} is a local basis ${\bf f}$ of
sections of $F$ such that $x\in U_{\bf f}$.  A {\it Riemannian metric} on
$F$ is a smooth section $G$ of the bundle ${\rm Hom}(F,F^*)$ such that for
each $x$ the bilinear form
$$
F(x)\times F(x)\ni (v,w)\to G_x(v,w)\equaldef\mylangle
G(x)v,w\myrangle\in\rr
$$
is symmetric and strictly positive definite.  If $G$ is a Riemannian
metric on $F$, then a local basis ${\bf f}=(f_1,\ldots,f_m)$ of
sections of $F$ is {\it orthonormal} if
$G_x(f_i(x),f_j(x))=\delta_{ij}$ for all $x\in U_{\bf f}$ and all
$i,j\in\{1,\ldots,m\}$, where $\delta_{ij}$ is Kronecker's delta.  It
is clear that, if $G$ is a Riemannian metric on the vector bundle $F$
over the manifold $M$, and $x\in M$, then a local orthonormal basis
of sections of $F$ near $x$ always exists, since it is always possible
to choose a local basis of sections of $F$ near $x$ and then construct
from it an orthonormal basis by means of the Gramm-Schmidt
orthonormalization procedure.


Suppose $M$ is a manifold.  We use $TM$, $T^*M$ to denote, respectively,
the tangent and cotangent bundles of $M$.  Then the expressions
$\pi_{_{TM}}$, $\pi_{_{T^*M}}$, $\Gamma(TM)$, $\Gamma(T^*M)$ denote,
respectively, the canonical projections from $TM$ and $T^*M$ to $M$, the
space of all smooth vector fields on $M$, and the space of all smooth
1-forms on $M$.  If $x\in M$, we write $T_xM $ and $T_x^*M$ ---rather than
$TM(x)$ and $T^*M(x)$--- to denote, respectively, the fibers of $TM$ and
$T^*M$ at $x$, i.~e. the tangent and cotangent spaces of $M$ at $x$.

\defi
A {\it singular subbundle} of a vector bundle $B$ on a manifold $M$ is a
pair $E=(F,\sigma)$, where $F$ is a vector bundle over $M$ and
$\sigma:\Gamma(F)\to\Gamma(B)$ is a homomorphism of
$C^\infty(M)$-modules.  \proofend


The following observation is trivial, but we prove it
in the appendix for completeness:
\xdef\propinduces{\the\sectno.\the\theono}
\prop 
If $(F,\sigma)$ is a singular subbundle of the vector bundle $B$ over $M$,
then $\sigma$ induces, for each $x\in M$, a linear map $\sigma_x:F(x)\to
B(x)$, such that, if $v\in F(x)$, and $V\in \Gamma(F)$ satisfies $V(x)=v$,
then $\sigma_x(v)=\sigma(V)(x)$. 
The maps $\sigma_x$ depend smoothly on $x$, in the sense that for each
$V\in\Gamma(F)$ the map $x\to \sigma_x(V(x))$ is smooth.
Conversely, any family $\{\sigma_x:x\in M\}$ of
linear maps $F(x)\to B(x)$ that depends smoothly on $x$ arises from a
module homomorphism $\sigma:\Gamma(F)\to\Gamma(B)$.  \proofend

If $E=(F,\sigma)$ is a singular subbundle of the bundle $B$ over $M$, and
$x\in M$, we write $E(x)=\sigma_x(F(x))$. We can then think of $E$ as a 
``vector bundle whose fibers are allowed to be of variable dimension.'' 


\medbreak If $(F,\sigma)$ is a singular subbundle of a vector bundle $B$ over
$M$, then Proposition \propinduces{} implies, in particular, that 
$\sigma(V)$ vanishes near $x$ if $V\in\Gamma(F)$ and $V\equiv 0$ near
$x$.  This implies that $\sigma(V)$ is a well defined member of
$\Gamma(B\restr\Omega)$ if $\Omega$ is an open subset of $M$ and
$V\in\Gamma(F\restr\Omega)$.

\defi
A {\it singular distribution} on a manifold
$M$ is a singular subbundle of the tangent bundle $TM$. \proofend

If $E=(F,\sigma)$ is a singular distribution on $M$, then 
the space  $\sigma(\Gamma(F))$ is a $C^\infty(M)$-submodule  of the Lie
algebra $\Gamma(TM)$ of all smooth vector fields on $M$. We use
$\Gamma(E)$ to denote this submodule,
and $\Lambda(E)$ to denote the Lie
subalgebra of $\Gamma(TM)$ generated by $\Gamma(E)$, and write
$\Lambda(E)(x)\equaldef\{X(x):X\in\Lambda(E)\}$.

If $E^i=(F^i,\sigma^i)$, $i=1,2$, are singular distributions on $M$, we can
define singular distributions $E^1+E^2$, $[E^1,E^2]$ as follows.  Let
$F=F^1\oplus F^2$ be the direct sum of the bundles $F^1$ and $F^2$, so the
fiber $F(x)$ is $F^1(x)\oplus F^2(x)$ for each $x$.  Let
$\sigma_x:F(x)\to T_xM$ be given by $\sigma_x(v_1,v_2)
=\sigma_x^1(v_1)+\sigma_x^2(v_2)$. Then $E^1+E^2$ is the singular
distribution $(F,\sigma)$. Let $\tilde F^i$, $i=1,2$, be the bundles of
$1$-jets of sections of $F^1$, $F^2$, and let $F'$ be the bundle
$\tilde F^1\otimes\tilde F^2$, so that 
$F'(x)=\tilde F^1(x)\otimes\tilde F^2(x)$ for each $x$.  If $v^1\in \tilde 
F^1(x)$, $v^2\in \tilde F^2(x)$, then we can find smooth sections $V^i$ of
$F^i$ such that the $1$-jet of $V^i$ at $x$ is $v^i$. The value
$[\sigma^1(V^1),\sigma^2(V^2)](x)$ of the Lie bracket 
$[\sigma^1(V^1),\sigma^2(V^2)]$ depends only on $v^1$ and
$v^2$, and is independent of the choice of the sections $V^1$, $V^2$. We
define $\tilde\sigma_x(v^1,v^2)=[\sigma^1(V^1),\sigma^2(V^2)](x)$. It
is then clear that
$\tilde\sigma_x:\tilde F^1(x)\times\tilde F^2(x)\to T_xM$ is a
bilinear map. Therefore $\tilde\sigma_x$ 
induces a linear map $\sigma_x':F'(x)\to T_xM$. We then define
$[E^1,E^2]$ to be the singular distribution $(F',\sigma')$.  It then follows
that the fiber $[E^1,E^2](x)$ is the linear span of all the brackets
$[X_1,X_2](x)$, for all $X_1\in\Gamma(E^1)$, $X_2\in\Gamma(E^2)$.  
Moreover, 
$$
E^1(x)\subseteq[E^1,E^2](x)\quad{\rm if}\quad E^2(x)\neq\{0\}\,.
$$
({\it Proof}:  Pick $Y\in\Gamma(E^2)$ such that $Y(x)\neq 0$. 
Let $v\in E^1(x)$.  Find $X\in\Gamma(E^1)$ such that 
$X(x)=v$. Find a function $\rho\in C^\infty(M)$ such that 
$\rho(x)=(Y\rho)(x)=1$. Then \hbox{$v\!=\![X,Y](x)\!-\![\rho X,Y](x)$}, so 
$v\!\in\![E^1,E^2](x)$.)
Similarly, $E^2(x)\!\subseteq\![E^1,E^2](x)$ if $E^1(x)\!\neq\!\{0\}$.

In particular, if $E$ is a singular distribution, 
we can define singular distributions $E^{k}$ inductively by letting
$$
E^{1}\,\,\equaldef\,\,E\,,\qquad
E^{k}\,\,\equaldef\,\,E+[E,E^{k-1}]\quad{\rm if}\quad k>1\,.
$$
If $E(x)\neq\{0\}$ for all $x$, then  $E^{k}$ is simply equal to
$[E,E^{k-1}]$ whenever $k>1$. 

\defi
If $k$ is a positive integer, and $E$ is a singular distribution on
the manifold $M$, we say that $E$ is {\it $k$-generating
at a point} $x$ if $E^k(x)=T_xM$, i.e. if 
$T_xM$ is the linear span of all the iterated brackets
$[X_1,[\ldots,[X_{i-1},X_i]\ldots]](x)$, $i\in\{1,\ldots,k\}$,
$X_1,\ldots,X_i\in\Gamma(E)$.  We say that $E$ is
{\it $k$-generating on a subset $S$ of $M$} if it is $k$-generating at 
every $x\in S$.  We call $E$ {\it bracket-generating} if
$\Lambda(E)(x)=T_xM$ for all $x\in M$.   \proofend

Equivalently, $E$ is bracket-generating iff for every $x$ there exists 
a positive integer $k$ such that $E$ is $k$-generating at $x$. 
For a subset $S$ of $M$, the smallest $k$ for which 
$E$ is $k$-generating on $S$ is called the {\it degree of nonholonomy}
of $E$ on $S$. Then $E$ is bracket-generating iff for every
compact subset $S$ of $M$ the degree of nonholonomy of $E$ on $S$ is
finite.

A {\it chart} of a manifold $M$ is a smooth diffeomorphism 
$\bx$ from an open subset ${\rm Dom}(\bx)$ of $M$ onto an open
subset ${\rm Im}(\bx)$ of $\rr^n$.  To a chart
$\bx=(x^1,\ldots,x^n)$ of $M$ we associate local bases of sections
$\partial^{\bx}$, $d\bx$ of $TM$, $T^*M$, with domain ${\rm
Dom}(\bx)$, by letting $\partial^{\bx}$ be the $n$-tuple
$(\partial_1^{\bx},\ldots,\partial_n^{\bx})$, where
$\partial_i^{\bx}$ is the vector field usually labeled
$\frac{\partial\,\,\,\,}{\partial{x^i}}$, and defining
$d\bx=(dx^1,\ldots,dx^n)$.  We also associate charts $\bx^{tan}$,
$\bx^{cot}$ of $TM$, $T^*M$, whose domains are the sets
$\pi_{_{TM}}^{-1}({\rm
Dom}(\bx))$, $\pi_{_{T^*M}}^{-1}({\rm{}Dom}(\bx))$, by letting
$\bx^{tan}=(x^1,\ldots,x^n,y^1,\ldots,y^n)$ and 
$\bx^{cot}=(x^1,\ldots,x^n,p_1,\ldots,p_n)$,
where the functions $y^i:\pi_{_{TM}}^{-1}({\rm Dom}(\bx))\to\rr$ and
$p_i:\pi_{_{T^*M}}^{-1}({\rm{}Dom}(\bx))\to\rr$ are given by
$y^i(x,y)=\mylangle{dx^i,y}\myrangle$, 
$\,p_i(x,p)=\mylangle p,\partial_i^{\bx}\myrangle$, 
and ``$x^i$'' really means $x^i\compo \pi_{_{TM}}$ or $x^i\compo
\pi_{_{T^*M}}$. 

The manifold $T^*M$ has a canonical symplectic form $\Omega_M$, given by
$\Omega_M=d\omega_M$, where $\omega_M(v)=\mylangle
p,d\pi_{_{T^*M}}(v)\myrangle$ whenever $(x,p)\in T^*M$, $v\in T_{(x,p)}T^*M$.
To every smooth function
$H\in C^\infty(T^*M)$ we associate the {\it Hamilton vector field}
$\vec{H}$ characterized by the identity
\xdef\identichar{\equanosectno}
$$
\Omega_M(X,\vec{H})=\mylangle dH,X\myrangle\quad{\rm for\,\,\,every}\quad
X\in\Gamma(T(T^*M))\,.\numeq
$$
To each vector field $g\in\Gamma(TM)$ we associate 
a function $\mu_g\in C^\infty(T^*M)$ ---the {\it momentum function} of $g$---
by letting 
\xdef\flamomen{\equanosectno}
$$
\mu_g(x,p)\,\equaldef\,\mylangle p,g(x)\myrangle\quad{\rm{}for}
\quad(x,p)\in T^*M\,.\numeq
$$
The vector field $\vec{\mu}_g$ is the {\it Hamiltonian lift} of $g$. It is
clear that  $d\pi_{_{T^*M}}(\vec{\mu}_g)=g$.

Suppose that  $\bx=(x^1,\ldots,x^n)$ is a chart of the manifold $M$
with domain $U$, and let 
$\bx^{cot}=(x^1,\ldots,x^n,p_1,\ldots,p_n):\pi_{_{T^*M}}^{-1}(U)\to\rr^{2n}$ 
be the corresponding chart of $T^*M$. Then the identities
$$
\omega_M=\sum_{i=1}^n p_i\,dx^i\,,\,\,\,\,\,
\Omega_M=\sum_{i=1}^n dp_i\wedge dx^i\,,
$$
hold on $\pi_{_{T^*M}}^{-1}(U)$.  Relative to $\bx^{cot}$, the Hamilton
vector field $\vec{H}$ corresponding to a function $H\in C^\infty(T^*M)$
has the expression
\xdef\expreham{\equanosectno}
$$
\vec{H}=\sum_i\Bigl(\frac{\partial H}{\partial{}p_i}
\frac{\partial\,\,}{\partial x^i}-\frac{\partial H}{\partial x^i}
\frac{\partial\,\,}{\partial{}p_i}\Bigr)\,,\numeq
$$
and the integral curves of $\vec{H}$ satisfy the system of $2n$ ordinary
differential equations 
\xdef\hamequas{\equanosectno}
$$
\dot x^i=\frac{\partial H}{\partial p_i}\,,\,\,\,\,\,\,
\quad\dot p_i=-\frac{\partial H}{\partial x^i}\,,\,\,\,\,\,\,
\quad i=1,\ldots,n\numeq
$$
known as  {\it Hamilton's equations}. It follows easily from (\hamequas) 
that
\xdef\hameqcons{\equanosectno}
$$
H\equiv{\rm constant}\quad{\rm along\,\,\,any\,\,\,trajectory\,\,\,of}
\quad\vec{H}\,.\numeq
$$


\defi
A {\it curve} in a topological space $X$ is a continuous map $\xi$ from a
nonempty interval $I(\xi)$ ---which is allowed to be open, closed or
half-open, and can be bounded or unbounded, and even trivial--- to $X$.  We
say that $\xi$ is an {\it arc in $M$} if $I(\xi)$ is compact.  We write
${\cal C}(M)$, ${\cal A}(M)$, to denote, respectively, the set of all
curves and the set of all arcs in $X$.  If 
$X$ is a manifold, we say that an arc $\xi$ in $X$ is an
{\it AC arc} if it is absolutely continuous.  We write
${\cal C}_{AC,loc}(X)$, ${\cal A}_{AC}(X)$ to
denote, respectively, the set of all locally
absolutely continuous curves in $X$ and the set of all AC
arcs in $M$. \proofend

If $M$  is a manifold, $\xi\in {\cal A}(M)$, and $I(\xi)=[a,b]$, then 
we write 
$$
\tau_-\xi\,\,\equaldef\,\,
a\,,\,\,\,\,\,\,\,\,\,\tau_+\xi\,\,\equaldef\,\, b
\,,\,\,\,\,\,\,\,\,\,\partial_-\xi\,\,\equaldef\,\,\xi(a)\,,\,\,\,\,\,\,\,\,\,
\partial_+\xi\,\,\equaldef\,\,\xi(b)\,,\,\,\,\,\,\,\,\,\,
\partial\xi\,\,\equaldef\,\,(\partial_-\xi,\partial_+\xi)\,,
$$
and refer to $\tau_-\xi$, $\tau_+\xi$, 
$\partial_-\xi$ and $\partial_+\xi$, respectively, as the 
{\it initial and terminal times} and the {\it initial and terminal 
points} of $\xi$, and to $\partial\xi$ as the {\it boundary value}, or 
{\it boundary condition}, or {\it endpoint condition}, 
of $\xi$.  If $\partial\xi=(\bar x,\hat x)$, we
say that $\xi$ {\it goes from $\bar x$ to $\hat x$}. 

\defi
If $E=(F,\sigma)$ is a singular distribution on the manifold $M$,
an {\it $E$-curve} is a curve $\xi\in {\cal C}_{AC,loc}(M)$ such that 
$\dot\xi(t)\in E(\xi(t))$ for almost every $t\in I(\xi)$. 
An {\it $E$-arc} is an $E$-curve which is an arc.  We use ${\cal C}_E(M)$,
${\cal A}_E(M)$ to denote, respectively, the set of all $E$-curves and the
set of all $E$-arcs in $M$. \proofend

If $\xi$ is an $E$-curve in $M$, then there exists a section
$\zeta$ of $F$ along $\xi$ ---i.~e. a map $I(\xi)\ni t\to \zeta(t)\in
F(\xi(t))$--- such that $\sigma_{\xi(t)}(\zeta(t))= \dot\xi(t)$ for
almost all $t\in I(\xi)$.  In general, $\zeta$ will not be unique, since the
maps $\sigma_x$ need not be injective.  It can be shown, however, that
$\zeta$ can always be chosen to be measurable:
\xdef\propismeasur{\the\sectno.\the\theono}
\prop
Let $E=(F,\sigma)$ be a smooth singular distribution on a manifold
$M$, and let $\xi$ be an $E$-curve. Then there exists a 
measurable map $I(\xi)\ni t\to \zeta(t)\in  F(\xi(t))$ such that 
$\dot\xi(t)=\sigma_{\xi(t)}(\zeta(t))$ for almost all $t\in
I(\xi)$. 

\beginproof This follows from
standard measurable selection theorems.  In addition, for our specific
situation there is a simple direct proof, as shown in the
appendix.\proofend

If $\xi$ is an $E$-curve in $M$, $J$ is a subinterval of 
$I(\xi)$, ${\bf f}=(f_1,\ldots,f_m)$ is a local
basis of sections of $F$ such that $\xi(J)\subseteq U_{{\bf f}}$, 
and $g_i=\sigma(f_i)$ for $i=1,\ldots,m$, then
Proposition \propismeasur{} tells us that there exist measurable functions
$\eta^i:J\to\rr$, $i=1,\ldots,m$, such that
\xdef\repre{\equanosectno}
$$
\dot\xi(t)=\sum_{i=1}^m\eta^i(t)g_i(\xi(t))\quad
{\rm for\,\,\,a.\,e.}\quad t\in J\,.\numeq$$

\defi
A vector-valued function $\eta=(\eta^1,\ldots,\eta^m)$ such that 
(\repre) holds is said to be 
a {\it local control representation} of
$\xi$ with respect to ${\bf f}$ on $J$.  
The functions $\eta^i$ are the {\it controls} corresponding to 
the representation $\eta$. \proofend

\xdef\itsuff{\the\sectno.\the\theono}
\rema
Local control representations are not unique, first 
because the local basis ${\bf f}$ is not unique, and second 
because, for a given ${\bf f}$, the functions $\eta^i$ need not be unique.
Once the local basis ${\bf f}$ is specified, choosing a control 
representation is of course tantamount to choosing a measurable map $\zeta$
such that $\zeta(t)\in F(\xi(t))$ and
$\sigma_{\xi(t)}(\zeta(t))=\dot\xi(t)$ for a.~e. $t$. 
\hbox{\phantom{aaaa}On the other hand, if
${\bf{}f}^\ell=(f_1^\ell,\ldots,f_m^\ell)$,  $\ell=1,2$, are} 
two local bases of sections, then $f_i^1=\sum_{j=1}^m\alpha_i^jf_j^2$ on
$U_{{\bf{}f}^1}\cap U_{{\bf f}^2}$, where the $\alpha_i^j$ are smooth
functions. So, if $J\subseteq I(\xi)$, $\xi(J)\subseteq U_{{\bf f}^1}\cap
U_{{\bf f}^2}$, and $\eta_1=(\eta_1^1,\ldots,\eta_1^m)$ is a
local control representation of $\xi$ on $J$
with respect to ${\bf f}^1$, then we can get a 
local control representation $\eta_2=(\eta_2^1,\ldots,\eta_2^m)$
of $\xi$ on $J$ with respect to ${\bf f}^2$ by letting 
$\eta_2^j=\sum_{i=1}^m\eta_1^i\alpha_i^j$, and it is then clear 
that $\eta_2$ is locally integrable, continuous, or smooth on $J$ if and
only if $\eta_1$ is. \proofend

\defi
We say that an $E$-curve $\xi$ on $M$ is {\it $E$-admissible} if, whenever
${\bf 
f}$ is a local basis of sections of $F$ and $J$ is a compact subinterval of
$I(\xi)$ such that $\xi(J)\subseteq U_{{\bf f}}$, it follows that $\xi$ has
an integrable local control representation $\eta$ on $J$ with respect to
${\bf f}$.  We use ${\cal C}_E^a(M)$, ${\cal A}_E^a(M)$ to denote,
respectively, the set of all $E$-admissible $E$-curves and the set of all
$E$-admissible $E$-arcs in $M$. \proofend

\rema
In view of Remark \itsuff{}, to prove that an $E$-curve $\xi$ is 
$E$-admissible it suffices to prove that for every 
$t\in I$ there exist a compact subinterval $J_t$ of $I(\xi)$ which is a
relative neighborhood of $t$ in $I(\xi)$, a
local basis ${\bf f}^t$ of sections of $F$ such that 
$\xi(J_t)\subseteq U_{{\bf f}^t}$, and an integrable 
local control representation $\eta_t$ on $J_t$ with respect to
${\bf f}^t$. \proofend


\rema If $E(x)$ has constant dimension, independent of $x$, then 
every $E$-curve is $E$-admissible. On the other hand, for general singular
distributions it is easy to give examples of absolutely continuous ---and
even smooth--- $E$-arcs that are not $E$-admissible (cf. Example A.1
in the appendix). \proofend

\xdef\propchow{\the\sectno.\the\theono}
\prop 
If a singular
distribution $E$ on $M$ is bracket-generating 
and $M$ is connected then for every $\bar x,\hat x\in M$
there exists a $\xi\in{\cal A}_{E}^a(M)$ such  that $\partial\xi=(\bar x,\hat
x)$. 

\beginproof This is a well-known consequence of ``Chow's Theorem,''
cf. Chow [\chow], Hermann [\herma],  Isidori [\isid], Jurdjevic [\jurdje],
Krener [\krener], Sussmann [\sussa]. \proofend 


\section{Subriemannian manifolds}
\xdef\secsub{\the\sectno}


\defi
A {\it subriemannian manifold} is a quadruple ${\cal
  M}=(M,F,\sigma,G)$ such  
that $M$ is a manifold, $(F,\sigma)$ is a bracket-generating smooth
singular distribution on $M$, and $G$  is a Riemannian metric on $F$.
The {\it dimension} of ${\cal M}$ is the dimension of $M$. \proofend

\mdno
{\sl From now on, we assume throughout the paper 
that ${\cal M}=(M,F,\sigma,G)$ is an $n$-dimensional subriemannian
manifold, and let $m={\rm dim}\,F$.  We write $E=(F,\sigma)$.}

\mdno
If ${\bf f}$ is a local basis of sections of $F$, then we write 
$U_{\bf f}^*\equaldef \pi_{_{T^*M}}^{-1}(U_{\bf f})$.  

\defi
If $x\in M$, then the {\it $G$-length} $||v||_{_G}$ of a vector $v\in
T_xM$ is defined to be the infimum of the numbers $\sqrt{\mylangle
  G(x)w,w\myrangle}$, ranging over all the vectors $w\in F(x)$ such that
$\sigma_x(w)=v$.  \proofend

Since the infimum of the empty set is equal to
$+\infty$, it is clear that 
$$
{\rm if}\quad v\in T_xM\,,\quad{\rm then}\quad
||v||_{_G}<\infty\,\,\Longleftrightarrow \,\,v\in E(x)\,.
$$

\xdef\remisnul{\the\sectno.\the\theono}
\rema  {\sl If $v\in E(x)$, then there exists a 
unique $w\in F(x)$ such that $\sigma_x(w)=v$ and 
$\mylangle G(x)w,w\myrangle=\|v\|_{_G}^2$.} (Indeed, the set $S(v)$ of those 
$w\in F(x)$ such that $\sigma_x(w)=v$ is an affine-linear subspace 
of $F(x)$, so the positive-definite quadratic form 
$w\to\mylangle G(x)w,w\myrangle$ has a unique minimum on $S(v)$.) It follows in
particular, that $||v||_{_G}=0$ iff $v=0$.   \proofend

We will use $\sigma_{x,c}^{-1}(v)$ to denote, for $v\in E(x)$, the
``canonical choice'' of a $w\in\sigma_{x}^{-1}(v)$ provided by Remark
\remisnul{}. That is, $\sigma_{x,c}^{-1}(v)$ is the unique $w\in F(x)$
such that $\sigma_x(w)=v$ and $\|w\|_{_G}=\|v\|_{_G}$.  

If $\xi\in{\cal C}_E(M)$, then $\sigma_{\xi(t),c}^{-1}(\dot\xi(t))$ 
is a well defined member of $F(\xi(t))$ for almost every $t\in I(\xi)$. 
We write 
\xdef\defome{\equanosectno}
$$
\omega_{\xi}(t)\,\,\,\equaldef\,\,\,\sigma_{\xi(t),c}^{-1}(\dot\xi(t))\,,\numeq
$$
so $\omega_{\xi}(t)\in F(\xi(t))$, $\|\omega_{\xi}(t)\|_{_G}=\|
\dot\xi(t)\|_{_G}$, and 
$\dot\xi(t)=\sigma_{\xi(t)}(\omega_{\xi}(t))$  
for almost every $t\in I(\xi)$. 

If $J$ is a subinterval $J$ of $I(\xi)$ and ${\bf f}$ is 
a local basis of sections  of $F$ such that 
$\xi(J)\subseteq U_{{\bf f}}$, then we can write 
$\omega_{\xi}(t)$ uniquely as $\sum_{i=1}^m\eta_{{\bf
f},\xi,J}^i(t)f_i(\xi(t))$ for a.~e. $t\in J$.  

\defi 
If $\xi$, $J$, ${\bf f}$ are as above, then the control 
$\eta_{{\bf f},\xi,J}$ is the {\it canonical control representation}
of $\xi$ on $J$ with respect to ${\bf f}$. \proofend

The following two elementary technical observations are proved in the
appendix:  
\xdef\propmeasmin{\the\sectno.\the\theono}
\xdef\propismea{\the\sectno.\the\theono}
\prop
If $\xi$ is a locally absolutely continuous curve in $M$, then the set
$I_0(\xi)\equaldef\Bigl\{t\!\!\in\!\!I(\xi)\!:\!||\dot\xi(t)||_{_G}\!\!<\!\!
\infty\Bigr\}$ is measurable, and the map $\omega_{\xi}$ is measurable on
$I_0(\xi)$.  In particular, (i)~the a.~e. defined function $I(\xi)\ni
t\to||\dot\xi(t)||_{_G}\in\rr\cup\{+\infty\}$ is measurable, 
(ii)~if $\xi$ is an $E$-curve, then $\omega_{\xi}$
is a.~e. defined and measurable on $I(\xi)$. \proofend

\xdef\propisad{\the\sectno.\the\theono}
\prop A locally absolutely continuous curve $\xi$ in $M$ is an
$E$-curve if and only if $||\dot\xi(t)||_{_G}<\infty$ for almost all
$t\in I(\xi)$.  Moreover, $\xi$ is $E$-admissible if and only if 
the function $t\to||\dot\xi(t)||_{_G}$ is locally integrable. \proofend

\defi
The {\it $G$-length} $||\xi||_{_G}$ of an absolutely continuous
arc $\xi$ in $M$ is the number 
$||\xi||_{_G}\,\,\equaldef\,\,\int_{I(\xi)}||\dot\xi(t)||_{_G}dt$, 
which is a well defined member of $[0,+\infty]$ by
Proposition \propismea. If $\xi$ is not absolutely continuous
we let $||\xi||_{_G}\,\equaldef\,+\infty$. \proofend

It follows from Proposition \propisad{} that

\xdef\propadmis{\the\sectno.\the\theono}
\prop
If $\xi\in{\cal{}A}(M)$, then $\xi$ is an $E$-admissible 
$E$-arc iff $||\xi||_{_G}<\infty$. \proofend

\defi
The {\it distance} $d_{{\cal M}}(\bar x,\hat x)$
between two points $\bar x$, $\hat x$ of $M$ is given by
$$
\eqalign{
d_{{\cal M}}(\bar x,\hat x)\,\,&\equaldef\,\,\, 
{\rm inf}\,\Bigl\{||\xi||_{_G}:\xi\in{\cal A}(M)\,\,\,,\,\,\,\,
\partial\xi=(\bar x,\hat x)\Bigr\}\cr
&=\,\,{\rm inf}\,\Bigl\{||\xi||_{_G}:\xi\in{\cal
A}_{E}(M)\,,\,\,\partial\xi=(\bar x,\hat x) \Bigr\}\cr
&=\,\,{\rm inf}\,\Bigl\{||\xi||_{_G}:\xi\in{\cal A}_{E}^a(M)\,,\,\,
\partial\xi=(\bar x,\hat x)\Bigr\}\,.\cr}
$$
(The equality of the the three right-hand sides follows easily
from Proposition \propadmis, together with the fact that 
${\rm{inf}}\,\emptyset=+\infty$.) \proofend

\rema {\it $(M,d_{{\cal M}})$ is a metric space, whose topology coincides
with the original manifold topology of $M$.}  This follows easily from the
fact that the definition of a subriemannian manifold used here includes the
requirement that $E$ be bracket-generating.  In particular, {\it the metric
space $(M,d_{{\cal M}})$ is locally compact.} \proofend

\rema If $M$ is connected then $d_{{\cal M}}(\bar x,\hat x)<+\infty$ for all
$\bar x,\hat x\in M$.  \proofend

We now state a well known estimate relating the subriemannian distance to
the distance arising from an ordinary Riemannian metric.  (For a proof,
cf. for example Gershkovich [\gershko,\gershkobb], Vershik-Gershkovich 
[\vershgea,\vershgeb].)

\xdef\propestimate{\the\sectno.\the\theono}
\prop
Let $g$ be a Riemannian  metric on $M$, and let $d_g$ be the corresponding
distance function.  Let $K$ be a compact subset of $M$, and let $k>0$ 
be an integer such that $E$ is $k$-generating on $K$.  Then there exist
positive constants $\Delta_1,\Delta_2$ such that the inequalities
$$
\Delta_1d_{g}(x_1,x_2)\le d_{{\cal M}}(x_1,x_2)\le
\Delta_2d_{g}(x_1,x_2)^{1/k}\numeq
$$
hold for all $x_1,x_2\in K$. \proofend

\defi
If $\xi_i:I_i\to M$, $i=1,2$, are arcs in $M$, then we say that
$\xi_1$ is a {\it time-reparametrization} of $\xi_2$ if there exists
an absolutely continuous monotonically nondecreasing function
$\theta:I_2\to I_1$ such that $\xi_1\compo\theta=\xi_2$.  We write
$TR(\xi)$ to denote the set of all time-reparametrizations of 
$\xi$. \proofend

\rema
The formula ``$\xi_1\in TR(\xi_2)$'' does not define an equivalence
relation, because (i)~the map $\theta$ need not be invertible, and
(ii)~even when $\theta$ is invertible, its compositional inverse need
not be absolutely continuous. \proofend

\defi
We use ${\cal C}^a({\cal M})$ (resp. 
${\cal A}^a({\cal M})$) as an alternative
notation for the set ${\cal C}_E^a(M)$ (resp. ${\cal A}_E^a(M)$) of
$E$-admissible $E$-curves  (resp. $E$-arcs) in $M$. If
$\xi\in{\cal C}^a({\cal M})$, then we say that $\xi$ is {\it parametrized
by a constant times arc-length} if there exists a constant $c$ such
that $||\dot\xi(t)||_{_G}=c$ for almost all $t\in I(\xi)$.  We 
say that $\xi$ is {\it parametrized by arc-length} if the constant 
$c$ is equal to $1$.    We call
$\xi$ {\it canonically parametrized} if it is parametrized by
constant times arc-length and $I(\xi)$ is the interval $[0,1]$.\proofend

\medbreak\noindent We use ${\cal C}^{PAL}({\cal M})$ (resp. ${\cal
C}^{PCAL}({\cal M})$, ${\cal C}^{CAN}({\cal M})$) to denote the set of all
curves $\xi\in{\cal C}^a({\cal M})$ that are parametrized by arc-length
(resp. parametrized by constant times arc-length, canonically
parametrized). We use ${\cal A}^{PAL}({\cal M})$, ${\cal A}^{PCAL}({\cal
M})$, ${\cal A}^{CAN}({\cal M})$ to denote the corresponding sets of arcs,
and write ${\cal A}^{PAL,0}({\cal M})$ to denote the set of all
$\xi\in{\cal A}^{PAL}({\cal M})$ such that $\tau_-\xi=0$.  Naturally,
${\cal C}^{CAN}({\cal M})={\cal A}^{CAN}({\cal M})$.

It is clear that 
\xdef\itiscle{\equanosectno}
$$
\xi\in{\cal A}^{PAL,0}({\cal M})\,\,\Longrightarrow\,\,
I(\xi)=[0,||\xi||_{_G}]\,,\numeq
$$
and 
\xdef\itiscli{\equanosectno}
$$
\xi\in{\cal A}^{CAN}({\cal M})\,\,\Longrightarrow\,\,
||\dot\xi(t)||_{_G}=||\xi||_{_G}\quad{\rm for\,\,\,a.~e.\quad}t\in[0,1]\,.\numeq
$$

The following two technical propositions are proved in the appendix.
\xdef\propinvarep{\the\sectno.\the\theono}
\prop
Arc-length is invariant under time-reparametrizations, in the
sense that, if $\xi_1$ and $\xi_2$ are locally absolutely continuous
arcs in $M$, and $\xi_1\in TR(\xi_2)$, then
$||\xi_2||_{_G}=||\xi_1||_{_G}$. \proofend

\xdef\propunique{\the\sectno.\the\theono}
\prop
If $\xi\in{\cal A}^a({\cal M})$, then the set
$TR(\xi)\cap{\cal{A}}^{PAL,0}({\cal M})$ consists of a unique
element $\xi_1$.
If $\xi$ is nontrivial (i.~e. if $I(\xi)$ is nontrivial), 
then there exists a unique
$\xi_2\in TR(\xi)\cap{\cal A}^{CAN}({\cal M})$. \proofend

\defi
If $\xi\in{\cal A}^a({\cal M})$, then the unique member of 
$TR(\xi)\cap{\cal{A}}^{PAL,0}({\cal M})$ 
is called the {\it arc-length reparametrization} of $\xi$.
If $\xi$ is nontrivial, then  the unique member 
of $TR(\xi)\cap{\cal A}^{CAN}({\cal M})$
is the {\it canonical reparametrization} of $\xi$. \proofend

\rema
If $\xi:\{a\}\to M$ is a trivial arc, then 
there is no ``canonical reparametrization'' in the sense of our definition,
but in that case we agree to call  the constant arc
$\xi_2:[0,1]\to M$ given by $\xi_2(t)=\xi(a)$ for $0\le t\le 1$
the {\it canonical reparametrization} of $\xi$. \proofend  

If $c\in[0,\infty\,[\,$, we use ${\cal C}^{a,\le c}({\cal M})$
---resp. ${\cal A}^{a,\le c}({\cal M})$--- to denote the set of all curves
---resp. arcs--- $\xi\in{\cal C}^a({\cal M})$ such that $||\dot\xi||_{_G}\le
c$ for almost all $t\in I(\xi)$.
We prove in the appendix that: 
\xdef\proplipimplies{\the\sectno.\the\theono}
\prop
If $\xi:[a,b]\to M$ is an arc, and $c>0$, then the following two 
conditions are equivalent: 
\itemitem{(a)} $d_{{\cal M}}(\xi(t_1),\xi(t_2))\le c|t_1-t_2|$
for all $t_1,t_2$ in $[a,b]$, 
\itemitem{(b)} $\xi\in{\cal A}^{a,\le c}({\cal M})$.\proofend

An important consequence of Proposition \proplipimplies{} is the following
\xdef\proplipimpliesb{\the\sectno.\the\theono}
\prop 
If $\,\,[a,b]$ $\,$is $\,$a $\,\,$compact $\,\,$interval, $\,\,$and $\,c\in\rr$,
$\,c>0$, $\,$then $\,$the $\,$set  
$C^0([a,b],M)\cap{\cal A}^{a,\le c}({\cal M})$ 
is closed in $C^0([a,b],M)$. 

\beginproof Since $d$ is continuous, the set of all arcs $\xi$ that satisfy 
Condition (a) of the statement of Proposition \proplipimplies{} is 
obviously closed. The conclusion then follows immediately  from 
Proposition \proplipimplies. \proofend

\defi A {\it length minimizer} of ${\cal M}$ is an arc 
$\xi\in{\cal{}A}^a({\cal M})$ with the property  that
$||\xi||_{_G}=d_{{\cal M}}(\partial\xi)$.  
A {\it local length minimizer} of ${\cal M}$ is an $E$-curve 
$\xi\in{\cal{}C}^a({\cal M})$ such that
for every $t\in I(\xi)$ there exists a compact subinterval 
$J$ of $I(\xi)$ with the property that $t$ belongs to the relative
interior of $J$ in $I(\xi)$ and the  restriction of $\xi$ to $J$ is a
length minimizer. \proofend

We use $Min({\cal M})$ ---resp. $LMin({\cal M})$--- to denote the set of
all length minimizers ---resp. local length minimizers--- of ${\cal{M}}$.
In addition, we use superscripts to denote various classes of members
of $Min({\cal M})$ or $LMin({\cal M})$, in a self-explanatory
way. For example, $LMin^{PAL}({\cal M})$ is the set of all local
length minimizers that are parametrized by arc-length, and
$Min^{CAN}({\cal M})$ is the set of all
canonically parametrized length minimizers. 

\rema
If $\xi$ is a length minimizer, then Proposition \propunique{} 
tells us that there exist $\xi_1\in TR(\xi)\cap Min^{PAL,0}({\cal M})$,
and $\xi_2\in TR(\xi)\cap Min^{CAN}({\cal M})$. Then 
$\partial\xi_1=\partial\xi_2=\partial\xi$, and Proposition 
\propinvarep{} implies that 
$||\xi_1||_{_G}=||\xi_2||_{_G}=||\xi||_{_G}$,  so $\xi_1$ and $\xi_2$ are
length minimizers as well. In particular, {\it if there exists
a length minimizer $\xi$ such that $\partial\xi=(\bar x,\hat x)$, 
then the set of all such minimizers contains at least one 
canonically parametrized arc, and one arc which is parametrized by arc-length
and defined on an interval of the form $[0,L]$.} \proofend

\xdef\isclosedprop{\the\sectno.\the\theono}
\prop
$Min^{CAN}({\cal M})$ is a closed subset of $C^0([0,1],M)$.

\beginproof Suppose $\{\xi_j\}_{j=1}^\infty$ is a sequence of members of 
$Min^{CAN}({\cal M})$ that converge uniformly to an arc $\xi:[0,1]\to M$. 
Let $L_j=||\xi_j||_{_G}$. Then $L_j=d_{{\cal M}}(\xi_j(0),\xi_j(1))$
since $\xi_j$ is a length minimizer. So $L_j\to 
L=d_{{\cal M}}(\xi(0),\xi(1))$ as $j\to\infty$. In particular, the sequence
$\{L_j\}$ is bounded. If $\Lambda_k=\sup\{L_j:j\in\nn,j\ge k\}$, then 
the fact that $\xi_j$ is canonically parametrized implies that 
$||\dot\xi_j(t)||_{_G}=L_j$ for almost all $t\in[0,1]$. Therefore
$d_{{\cal M}}(\xi_j(t),\xi_j(t'))\le\Lambda_k|t-t'|$ whenever 
$t,t'\in[0,1]$ and $j\ge k$. So, for every $k$,  
$d_{{\cal M}}(\xi(t),\xi(t'))\le\Lambda_k|t-t'|$ whenever 
$t,t'\in[0,1]$. Therefore 
$d_{{\cal M}}(\xi(t),\xi(t'))\le L|t-t'|$ whenever 
$t,t'\in[0,1]$. Proposition \proplipimplies{} then implies that 
$\xi\in{\cal A}^{a,\le L}({\cal M})$. So $||\dot\xi(t)||_{_G}\le L$ 
for almost all $t$, and then 
$L\,\,=\,\,d_{{\cal M}}(\xi(0),\xi(1))\,\,\le\,\,
||\xi||_{_G}\,\,=\,\,\int_0^1||\dot\xi(t)||_{_G}dt\,\,\le\,\,L$.
Therefore $||\xi||_{_G}=d_{{\cal M}}(\xi(0),\xi(1))$, showing that $\xi$ is a
minimizer, and $\int_0^1||\dot\xi(t)||_{_G}dt\le L$, so 
$||\dot\xi(t)||_{_G}=L$ for almost all $t$, and then 
$\xi$ is canonically parametrized. \proofend

If $a,b\in\rr$ and $a\le b$, we use
${\cal{}A}_{[a,b]}({\cal{}M})$ to denote the set of all absolutely
continuous arcs in $M$ whose interval of definition is $[a,b]$. 
\defi
The extended-real-valued functional ${\cal E}_{{\cal M},a,b}:{\cal
A}_{[a,b]}({\cal{}M})\to\rr\cup\{\infty\}$ given by
${\cal E}_{{\cal M},a,b}(\xi)=\int_a^b||\dot\xi(t)||_{_G}^2\,dt$
is the {\it energy functional} of ${\cal M}$ on the interval
$[a,b]$. \proofend 

\noindent(It is clear that the above integral is defined
---but need not be finite--- for all absolutely continuous arcs
$\xi:[a,b]\to M$.) We let 
\xdef\isdefomin{\equanosectno}
$$
d^{\,{\cal E},a,b}(\bar x,\hat x)=\inf\Big\{{\cal E}_{{\cal M},a,b}(\xi):
\xi\in{\cal A}_{[a,b]}({\cal{}M})\,,\,\partial\xi=(\bar x,\hat
x)\Big\}\,.\numeq 
$$

\defi
An arc $\xi\in{\cal A}_{[a,b]}({\cal{}M})$ is 
an {\it energy minimizer} if 
${\cal E}_{{\cal M},a,b}(\xi)=d^{\,{\cal E},a,b}(\partial\xi)$. 
A {\it local energy minimizer} of ${\cal M}$ is an $E$-curve 
$\xi\in{\cal{}C}^a({\cal M})$ such that
for every $t\in I(\xi)$ there exists a compact subinterval 
$J$ of $I(\xi)$ with the property that $t$ belongs to the relative
interior of $J$ in $I(\xi)$ and the  restriction of $\xi$ to $J$ is an
energy minimizer. \proofend

The following proposition is proved in the appendix:
\xdef\equivlengen{\the\sectno.\the\theono}
\prop
If $a,b\in\rr$, $a<b$, then:

{\parindent=30pt

{\sl

\item{(A)} $d^{\,{\cal E},a,b}(\bar x,\hat x)=(b-a).d_{{\cal M}}(\bar x,\hat
  x)^2$ for all $\bar x,\hat x$ in $M$; 

\item{(B)} if $\xi\in{\cal A}_{[a,b]}({\cal{}M})$, then 
the following are equivalent: 

\medbreak
\itemitem{(B.1)} $\xi$ is an energy minimizer;

\itemitem{(B.2)} $\xi$ is a length minimizer parametrized by constant times
arc-length. \proofend
}

}

\xdef\remonriem{\the\sectno.\the\theono}
\rema
{\it (The Riemannian case).} A Riemannian manifold is a subriemannian
manifold ${\cal M}=(M,F,\sigma,G)$ such that $F=TM$ and $\sigma$ is the
identity map.  More generally, if a subriemannian manifold ${\cal
M}=(M,F,\sigma,G)$ is such that $\sigma$ is surjective ---i.~e.,
equivalently, $\sigma_x$ is surjective for each $x\in M$--- then there is a
Riemannian metric $G^{\#}$ on $M$ that gives rise to the same length and
energy functionals, and the same distance function.  Indeed, $F$ splits as
the direct sum $F_1\oplus F_2$, where $F_2$ is the kernel of $\sigma$, and
$F_1$ is such that $\sigma_1=\sigma\restr F_1$ is a bundle isomorphism onto
$TM$. Then the restriction $G_1$ of the metric $G$ to $F_1$ induces, via
the isomorphism $\sigma_1$, an ordinary Riemannian metric $G^{\#}$ on
$TM$.  It is then clear that in this case every locally absolutely continuous
curve in $M$ is an $E$-admissible $E$-curve.  Moreover, the definition of
the $G$-length $||v||_{_G}$ of a tangent vector is such that $||v||_{_G}$
is precisely the Riemannian length $||v||_{_{G^{\#}}}$. \proofend


\section{Completeness,  convexity, and the Hopf-Rinow theorem}

\xdef\extenproper{\the\sectno.\the\theono}
\defi
Let ${\cal C}$ be a set of curves in a manifold $M$. A member $\xi$
of ${\cal C}$ is {\it maximal within ${\cal C}$} if there does not
exist a $\xi'\in{\cal C}$ such that $I(\xi)\subseteq I(\xi')$,
$I(\xi)\neq I(\xi')$, and $\xi'(t)=\xi(t)$ for all $t\in I(\xi)$.
We use $\maxi{{\cal C}}$ to denote the set of all 
$\xi\in{\cal C}$ that are maximal within ${\cal C}$.
We say that ${\cal C}$ has the {\it extension property} if,
whenever $\xi\in\maxi{{\cal C}}$ and $\tau$ is a limit point 
of $I(\xi)$ such that $\lim_{t\to_{_{I(\xi)}}\tau}\xi(t)$ exists, 
it follows that $\tau$ is an interior point of $I(\xi)$. \proofend



\xdef\extexamp{\the\sectno.\the\theono}
\exam
{\it The class
${\cal{}C}^{PAL}({\cal M})$ of all admissible curves parametrized by
arc-length has the extension property.
If $c>0$, then the class ${\cal{}C}^{a,\le c}({\cal M})$ also has the
extension property.} (See the appendix for a proof.) \proofend


\exam
{\it In a riemannian manifold ${\cal M}=(M,G)$, the class
${\cal{}G}^{PAL}({\cal M})$ of all geodesic arcs parametrized by
arc-length has the extension property.} This follows easily 
from the fact that an arc $\xi$ is in ${\cal{}G}^{PAL}({\cal M})$ iff
it is parametrized by arc-length and is a solution of the geodesic
equation.  We will see below that a similar conclusion is true for strongly 
bracket-generating subriemannian manifolds. \proofend

\defi
Let $\bar\xi=\{\xi_j\}_{j=1}^\infty$ be a sequence of curves in a metric
space $X$, and let 
$\xi$ be a curve in $X$. We say that $\bar\xi$ {\it converges to $\xi$ on
compact intervals} if, whenever $J$ is a compact subinterval of $I(\xi)$,
it follows that $J\subseteq I(\xi_{j})$ for sufficiently large $j$, and
the $\xi_j$ converge uniformly to $\xi$ on $J$. \proofend

We will need the following fact ---proved in the
appendix--- which is a special case of a general result on ``arc
systems,'' namely, Proposition 2 of \S 2 of Liu-Sussmann [\liuss{}]: 

\xdef\compaprop{\the\sectno.\the\theono}
\prop
If $c\!\!>\!\!0$, then ${\cal C}^{a,\le c}({\cal M})$ has the following
compactness property: 

{\it 

{\parindent=24pt

\item{(C)} If $\{\xi_j\}_{j=1}^\infty$ is a sequence in
$\maxi{{\cal C}^{a,\le  c}({\cal M})}$  
and $\{t_j\}_{j=1}^\infty$ is a sequence in $\rr$ such that 
$t_j\in I(\xi_j)$ for each $j$, and the limits
$\bar t=\lim_{j\to\infty}t_j$ and $\bar x=\lim_{j\to\infty}\xi_j(t_j)$
exist, then there exist a subsequence $\{\xi_{j(k)}\}_{k=1}^\infty$ 
of $\,\,\{\xi_j\}_{j=1}^\infty$ and a curve
$\xi\in\maxi{{\cal C}^{a,\le c}({\cal M})}$ such that
$\,\,\bar t\in I(\xi)$ and $\{\xi_{j(k)}\}_{k=1}^\infty$ 
converges to $\xi$ on compact intervals. \proofend

}
}


\medbreak
\defi We call ${\cal M}$ {\it complete} if the
metric space $(M,d_{{\cal M}})$ is complete. \proofend

We define  
\xdef\defball{\equanosectno}
$$
B_{{\cal M}}(x,r)\,\,\equaldef\,\,\Big\{x'\in M:d_{{\cal
M}}(x',x)\le r\Big\}\,.\numeq
$$ 
The following statement is then the subriemannian analogue of the
well-known Hopf-Rinow theorem of Riemannian geometry.  The proof is given
in the appendix.  

\xdef\hopfrinow{\the\sectno.\the\theono}
\theo
The following four conditions are equivalent: 

{\sl

{\parindent=25pt

\item{(i)\phantom{i}\phantom{i}} ${\cal M}$ is complete;

\item{(ii)\phantom{i}} every closed ball $B_{{\cal M}}(x,r)$ is
compact; 

\item{(iii)} if $\{x_j\}_{j=1}^\infty$ is any sequence of 
points of $M$ whose distances to a given point $\bar x$ are bounded above
by a finite constant, then 
there exist a subsequence of $\{x_j\}_{j=1}^\infty$ that converges to
a limit $x$, and a length minimizer $\xi$ such that
$\partial\xi=(\bar x, x)$;

\item{(iv)} every curve $\xi\in\maxi{{\cal C}^{a,\le 1}({\cal M})}$ 
satisfies $I(\xi)=\rr$;

\medbreak\noindent If the class $LMin^{PAL}({\cal M})$ of arc-length
parametrized locally length-minimizing curves of ${\cal M}$ has the
extension property, then (i), (ii), (iii) and (iv) are equivalent to

\medbreak

\item{(v)} every $\xi\in\maxi{LMin^{PAL}({\cal M})}$ 
satisfies $I(\xi)=\rr$. \proofend

} 

}

\medbreak
The following proposition is also proved in the appendix:

\xdef\eximinloc{\the\sectno.\the\theono}
\prop 
If $\bar x,\hat x\in M$, $r\ge 0$, are such that $d_{{\cal M}}(\bar x,\hat
x)\le r$ and $B_{{\cal M}}(\hat x,r)$ is compact, then there exists a
length minimizer $\xi$ such that $\partial\xi=(\bar x,\hat x)$. \proofend

\xdef\relcondef{\the\sectno.\the\theono}
\defi
Let $S$ be a subset of $M$. We call $S$ {\it relatively convex in ${\cal
    M}$} if any two points of $S$ can be joined by a length minimizer 
$\xi$ in ${\cal M}$.        \proofend

\theo
Every point of $M$ has a neighborhood which is
relatively convex in ${\cal M}$.  If ${\cal M}$ is complete and connected,
then $M$ is relatively convex in ${\cal M}$.

\beginproof
The first assertion follows from Proposition \eximinloc: given $x\in
M$, we can always find $r>0$ such that 
$B_{{\cal M}}(x,r)$ is compact; then, if $\bar x$, $\hat x$ belong to 
$U=B_{{\cal M}}(x,{r\over 2})$, the ball 
$B_{{\cal M}}(\bar x,{r\over 2})$ is closed and entirely contained in 
$B_{{\cal M}}(x,r)$. So 
$B_{{\cal M}}(\bar x,{r\over 2})$ is compact. Therefore there exists a
length minimizer joining $\bar x$ to $\hat x$. 

The second assertion follows from Theorem \hopfrinow: if ${\cal M}$ is
complete, then (iii) of that theorem holds; given $\bar x$, $\hat x$,
we can choose $x_j=\hat x$ for all $j$, and then (iii) yields a
length minimizer $\xi$ such that $\partial\xi=(\bar x,\hat x)$. \proofend

\section{Extremals and geodesics}

\xdef\secextre{\the\sectno}

\noindent The {\it Pontryagin Maximum Principle} (cf. Pontryagin {\it
et al.} [\pontr], Berkovitz [\berko], Lee-Markus~[\leemark] for the Maximum
Principle in $\rr^n$, Sussmann [\susscofree] for coordinate-free versions
on manifolds, Strichartz [\stricha,\strichb] and Liu-Sussmann [\liussmem]
for the application to subriemannian minimizers) gives a necessary
condition for a curve $\xi:[a,b]\to M$ to be an energy minimizer.  The
condition says that $\xi$ has to be an ``extremal,''  in a sense that we
now define. 

\defi
If $\xi$ is a curve in a manifold $M$, then a {\it cotangent lift}  of
$\xi$ is a curve $\Xi:I(\xi)\to T^*M$ such that 
$\pi_{_{T^*M}}\compo\Xi=\xi$. \proofend

Clearly, a cotangent lift $\Xi$ of a curve $\xi$ is given by 
$\Xi(t)=(\xi(t),\psi(t))\quad{\rm for}\quad t\in I(\xi)$, 
where $\psi$ is a {\it field of covectors along $\xi$}, i.~e. a
map such that $\psi(t)\in T_{\xi(t)}^*M$ for each $t\in I(\xi)$. 

\xdef\defofextremal{\the\sectno.\the\theono}
\defi
An {\it extremal} is an $E$-admissible $E$-curve $\xi$
such that there exist a locally absolutely continuous cotangent lift 
$\Xi:I(\xi)\to T^*M$ and a constant $\psi_0$ that satisfy, if
$\Xi(t)=(\xi(t),\psi(t))$,  the conditions:

{\it

\itemitem{(E.1)}  $\psi_0\ge 0$, and $(\psi(t),\psi_0)\neq (0,0)$ for all
$t\in I(\xi)$, 

\itemitem{(E.2)} for almost every $t\in I(\xi)$, the function 
$F\Big(\xi(t)\Big)\ni v\to \Big\langle\psi(t),\sigma_{\xi(t)}(v)\Big\rangle-
{\psi_0\over 2}\Bigr\|v\Bigl\|^2$
is maximized by the value $v=\omega_{\xi}(t)$; 

\itemitem{(E.3)} for every local orthonormal basis ${\bf{}f}=
(f_1,\ldots,f_m)$ of sections of $F$, every coordinate chart
$\bx=(x^1,\ldots,x^n)$ of $M$, and every compact subinterval $[a,b]$ 
of $I(\xi)$ with the property that 
$\xi([a,b])\subseteq U_{{\bf f}}\,\cap\,{\rm Dom}(\bx)$, 
if we write   $g_i=\sigma(f_i)$, then  the field of
covectors $\psi$ satisfies the ``adjoint equation''
\xdef\adjequa{\equanosectno}
$$
\dot\psi(t)=-\sum_{i=1}^m\eta_{{\bf f},\xi,[a,b]}^i(t).\psi(t).{\partial
  g_i\over\partial x}(\xi(t))\quad{\rm for\,\,\,a.~e.}\quad t\in[a,b]\,,\numeq
$$
relative to the chart $\bx$. \proofend

}

\medbreak\noindent As stated, Condition (E.3) of Definition \defofextremal{}
appears to be highly nonintrinsic, because it seems to depend on the
choice of ${\bf f}$ and $\bx$. This, however, is not so, as we now
show. 

To state (E.3) in an intrinsic way, we first introduce the vector bundle
$\hat F$ over $T^*M$ obtained by pulling back $F$ via the map
$\pi_{_{T^*M}}:T^*M\to M$.  That is, $\hat F$ is the bundle over $T^*M$
whose fiber $\hat F(x,p)$ over a point $(x,p)\in T^*M$ is just $F(x)$.  The
points of $\hat F$ are then the triples $(x,p,v)$ such that $x\in M$, $p\in
T_x^*M$, and $v\in F(x)$.

We then define the   ``control Hamiltonian''
$H_{{\cal{M}}}^c:\hat{F}\times\rr\to\rr$ by letting   
\xdef\contham{\equanosectno}
$$
H_{{\cal M}}^c(x,p,v,p_0)\,\,\equaldef\,\,\Big\langle
p,\sigma_x(v)\Big\rangle-{p_0\over 2}||v||_{_G}^2\,. \numeq
$$
The {\it critical set} $\,\,Crit({\cal M})$ is then the set 
of all $(x,p,v,p_0)\in\hat F\times\rr$ that satisfy the equation
${\partial H_{{\cal M}}^c\over\partial v\,\,\,\,\,}(x,p,v,p_0)=0$.
Then $\,\,Crit({\cal M})$ is an intrinsic object, 
because, for fixed $(x,p,p_0)$, $v$ varies in the linear space 
$F(x)$, so the gradient of $H_{{\cal M}}^c$ with respect to 
$v$ is well defined. 

Now, if $(x,p,v,p_0)\in Crit({\cal M})$, we define a tangent
vector $\vec{H}_{{\cal M}}^c(x,p,v,p_0)$ as follows:  

{\parindent=50pt

\medbreak
\item{{\it (Step 1)}} We choose, in an arbitrary fashion, a smooth section $V$ of
$\hat F$. 

\item{{\it (Step 2)}} Define a real-valued function 
$H_{{\cal{}M}}^{c,V,p_0}$ on the cotangent bundle $T^*M$, by  letting 
$H_{{\cal M}}^{c,V,p_0}(x',p')=H_{{\cal M}}^{c}(x',p',V(x',p'),p_0)$
for $(x',p')\in T^*M$.

\item{{\it (Step 3)}} By definition, then, 
\xdef\thevec{\equanosectno}
$$
\vec{H}_{{\cal M}}^c(x,p,v,p_0)\,\equaldef\,\vec{H}_{{\cal
    M}}^{c,V,p_0}(x,p)\,\numeq
$$
where $\vec{H}_{{\cal M}}^{c,V,p_0}$ is the Hamiltonian vector field
arising from the function $H_{{\cal M}}^{c,V,p_0}$.

}


\medbreak\noindent To justify the definition, we need the following fact,
proved in the appendix:

\medbreak\item{($*$)} {\it the vector 
$\vec{H}_{{\cal M}}^c(x,p,v,p_0)\in T_{x,p}T^*M$ defined by (\thevec) 
does not depend on the choice of the section $V$.}

\medbreak The following proposition ---proved in the appendix--- then 
tells us that (E.3) is an intrinsic condition provided that (E.2) holds:

\xdef\efintrinsic{\the\sectno.\the\theono}
\prop
Let $\xi$ be an $E$-admissible $E$-curve. Let $\psi:I(\xi)\to T^*M$ be a
field of covectors 
along $\xi$ and let $\psi_0$ be a constant such that, 
if $\,\,\Xi(t)=(\xi(t),\psi(t))$, then Condition (E.2) of Definition
\defofextremal{} holds.  Then:  

{\sl

\item{(A)} $(\xi(t),\psi(t),\omega_{\xi}(t),\psi_0)\in Crit({\cal M})$ for
almost every $t\in I(\xi)$;

\item{(B)}  (E.3) holds if and only if 
$\dot\Xi(t)=\vec{H}_{{\cal M}}^c(\Xi(t),\omega_{\xi}(t),\psi_0)$ for a.~e.
$t\in I(\xi)$.\proofend

}

The maximum principle now implies the following necessary condition 
for an arc to be an energy minimizer. The details of the derivation are given in
the appendix. 

\xdef\minisareopt{\the\sectno.\the\theono}
\theo
If $\xi_*:[a,b]\to M$ is an energy minimizer, then $\xi_*$ is an
extremal. \proofend

\defi 
A {\it normal extremal} of ${\cal M}$ is an extremal that satisfies the
conditions of Definition \defofextremal{} with a nonvanishing value of
$\psi_0$.  An {\it abnormal extremal} of ${\cal M}$ is an extremal that
satisfies the conditions of Definition \defofextremal{} with $\psi_0=0$.
\proofend

\rema
An extremal $\xi$ can be both normal and abnormal, since it is possible for
$\xi$ to satisfy the conditions of Definition \defofextremal{} with
different choices of the pair $(\psi,\psi_0)$ and, in particular, it can
happen that $\psi_0=0$ for some of these pairs but not for all of
them.\proofend 

\rema
{\it A normal extremal satisfies the conditions of Definition
\defofextremal{} with $\psi_0=1$}.  Indeed, it is easy to see that, if a
pair $(\psi,\psi_0)$ is such that these conditions hold, and
$r\in\rr$ $r>0$, then the pair $(r\psi,r\psi_0)$ also satisfies
the conditions. \proofend

It turns out that the normal extremals are projections on $M$ of curves
$\Xi$ in  $T^*M$ that satisfy a Hamiltonian equation called the ``geodesic
equation,'' as we now explain.


\medbreak If $p\in T_x^*M$, then the composite map
$p\compo\sigma_x:F(x)\to\rr$ is a linear functional on the inner
product space $F(x)$, so $p\compo\sigma_x$ has a well-defined operator 
norm $||p||_{_G}$, given by the formula
$||p||_{_G}={\rm sup}\Big\{\mylangle p,\sigma_x(v)\myrangle: v\in
F(x),||v||_{_G}\le 1\Big\}$. 
\defi
The real-valued function $H_{{\cal{}M}}$ on $T^*M$ given by
$H_{{\cal{}M}}(x,p)={1\over 2}||p||_{_G}^2$ for $(x,p)\in T^*M$ 
is the {\it Hamiltonian} of ${\cal M}$. The integral curves of
$\vec{H}_{{\cal{}M}}$ are the {\it solutions of the geodesic 
equation} of ${\cal M}$. The curves on $M$ that are projections of
solutions of the geodesic equation are the {\it geodesics}.  We write
$GEO({\cal M})$, $\widehat{GEO}({\cal M})$ to denote, respectively, the set
of all geodesics and the set of all solutions of the geodesic
equation of ${\cal M}$.  \proofend

It follows immediately from (\hameqcons) that
$$
H_{{\cal{}M}}\equiv{\rm constant}\,\,\,{\rm
along\,\,\,every\,\,\,}\Xi\in\widehat{GEO}({\cal M}).\numeq
$$

If ${\bf f}=(f_1,\ldots,f_m)$ is a local orthonormal local basis of
sections of $F$, and $(x,p)\in U_{\bf f}^*$, then
$||p||_{_G}$ is given by the formula 
$||p||_{_G}^2= \sum_{i=1}^m\mylangle p,g_i(x)\myrangle^2$, 
where  $g_i=\sigma(f_i)$. Therefore
\xdef\expre{\equanosectno}
$$H_{{\cal{}M}}(x,p)=\frac{1}{2}\sum_{i=1}^m\mylangle p,g_i(x)\myrangle^2\,.\numeq$$
Equivalently,  if we use $\mu_X$ to denote, for a vector field $X$ on $M$, 
the momentum function $(x,p)\to\mu_X(x,p)\equaldef\mylangle
p,X(x)\myrangle$, defined earlier (cf. Formula (\flamomen)), then 
\xdef\expretwo{\equanosectno}
$$
H_{{\cal{}M}}=\frac{1}{2}\sum_{i=1}^m(\mu_{g_i})^2\,\,\,{\rm on}\,\,\,
U_{\bf{}f}^*. \numeq
$$
It is clear that, relative to the chart $\bx^{cot}$ of $T^*M$ induced by a 
coordinate chart $\bx$ of $M$, 
$$
{\partial H_{{\cal{}M}}\over\partial p\,\,\,\,\,\,\,}(x,p)=
\sum_{i=1}^m\mylangle p,g_i(x)\myrangle.g_i(x)\,\,\,\,\,\,{\rm and}
\,\,\,\,\,\,
{\partial H_{{\cal{}M}}\over\partial x\,\,\,\,\,\,\,}(x,p)=
\sum_{i=1}^m\mylangle p,g_i(x)\myrangle.p.
{\partial g_i\over\partial x\,}(x)\,,\numeq
$$
where we are treating $p$ as a row vector, and  each
${\partial g_i\over\partial x\,}(x)$ is a square matrix.

Then the geodesic equation, in coordinates, becomes the system:
\xdef\geosystem{\equanosectno}
$$
\dot x=\sum_{i=1}^m\mylangle p,g_i(x)\myrangle.g_i(x)\,,\,\,\,\,\,\,\,\,\,\,\,\,
\dot p=-\sum_{i=1}^m\mylangle p,g_i(x)\myrangle.p.
{\partial  g_i\over\partial x\,}(x)\,.\numeq
$$
This is a smooth system of first order ordinary differential
equations on $T^*M$, so 

\prop
Every solution of the geodesic equation is smooth. \proofend

\xdef\krone{\the\sectno.\the\theono}
\rema
The geodesic equations (\geosystem) should really have been written in
the form  
\hbox{$\dot x=\sum_{i,j=1}^m\delta^{ij}\mylangle p,g_i(x)\myrangle.g_j(x)$, 
$\dot p=-\sum_{i,j=1}^m\delta^{ij}\mylangle p,g_i(x)\myrangle.p.
{\partial  g_i\over\partial x\,}(x)$} ---where $\delta^{ij}$ is Kronecker's
delta--- so as to preserve the usual summation conventions. \proofend 

Suppose that $\Xi\in\widehat{GEO}({\cal M})$, and write
$\Xi(t)=(\xi(t),\psi(t))$. Then $\xi$ is an $E$-admissible $E$-curve,
having a control representation with control $\eta$ given by
$$
\eta(t)=\Big(\eta^1(t),\ldots,\eta^m(t)\Big)=
\tilde\ffi_{{\bf f}}\Big(\Xi(t)\Big)=\Big(\tilde\ffi_{{\bf f}}^1
\Big(\Xi(t)\Big),\ldots,\tilde\ffi_{{\bf f}}^m\Big(\Xi(t)\Big)\Big)\,,
$$
where $\tilde\ffi_{{\bf f}}^i=\ffi_{{\bf f},i}$ (that is, 
$\tilde\ffi_{{\bf f}}^i=\sum_{j=1}^m\delta^{ij}\ffi_{{\bf f},j}$,
cf. Remark \krone), and 
the $\ffi_{{\bf f},i}$ are just the momentum functions
$\mu_{g_i}$, so that  
\xdef\defofffi{\equanosectno}
$$
\ffi_{{\bf f},i}(x,p)\,\,\equaldef\,\,\mylangle p,g_i(x)\myrangle\,.\numeq
$$
We prove in the appendix that 
\xdef\iscanonical{\the\sectno.\the\theono}
\prop
If $\bx$ is a chart of $M$, ${\bf f}$ is a local orthonormal basis of
sections of $F$, 
\ \hbox{$\xi\in{\cal C}_{AC,loc}(M)$, 
\ $I(\xi)\subseteq U_{\bf f}\,\cap\,{\rm Dom}(\bx)$}, 
\ $\psi$ is a field of covectors 
along $\xi$, and \hbox{$\dot\xi(t)=\sum_{i=1}^m\tilde\ffi_{{\bf
f}}^i(\xi(t),\psi(t))g_i(\xi(t))$ a.~e.,} then the control
$t\to\tilde\ffi_{{\bf f}}(\xi(t),\psi(t))$ is the canonical control
representation $\eta_{{\bf f},\xi,I(\xi)}$ of $\xi$ with respect to ${\bf
f}$ on $I(\xi)$. \proofend

\noindent The field of covectors $\psi$ is then a solution of 
$\dot\psi(t)=\,-\,\psi(t).\sum_{i=1}^m\eta_{{\bf
f},\xi,I(\xi)}^i(t).{\partial g_i\over\partial x\,}(\xi(t))$, which is
precisely the adjoint equation (\adjequa). So, if we let $\psi_0\equiv 1$,
then Conditions (E.1) and (E.3) of Definition \defofextremal{} are
satisfied.  Moreover, it is clear that the quadratic form
\ $\rr^m\ni (u^1,\ldots,u^m)\to\sum_{i=1}^mu^i\mylangle p,g_i(\xi(t))\myrangle
-{\psi_0\over 2}||u||^2\in\rr$ \ is maximized for the values 
\hbox{$u^i=\mylangle p,g_i(\xi(t))\myrangle=\eta_{{\bf f},\xi,I(\xi)}^i(t)$}
of the $u$ variables, so (E.2) holds as well.

Conversely, suppose $\xi$ is a normal extremal, and let $\psi$ be such that
Conditions (E.1,2,3) of Definition \defofextremal{} hold with $\psi_0=1$.
Write $\Xi(t)=(\xi(t),\psi(t))$.  Then the minimization condition implies
that $\eta_{{\bf f},\xi,I(\xi)}(t)=\tilde\ffi_{\bf f}(\Xi(t))$, so $\Xi$ is a
solution of the geodesic equation, and $\xi$ is a geodesic. We have
therefore proved:
\prop
A curve $\xi$ is a geodesic if and only if it is a normal extremal. \proofend

So far, we have found a necessary condition for an arc to be a minimizer,
namely, that it be an extremal.  This condition is not in general
sufficient, even for local minimization, but the following weaker fact is
true: 

\xdef\geoislocmin{\the\sectno.\the\theono} 
\theo
Every geodesic curve is a local energy minimizer.  

\beginproof This is a standard fact in optimal control theory, proved by
the technique of ``fields of extremals,'' i.~e. by using the method of
characteristics to solve the Hamilton-Jacobi equation.  For a detailed
proof, see Hamenst\"adt [\hamen], or Liu-Sussmann [\liussmem], where the
proof, although it is only given for the case of a nonsingular distribution
$E$, works equally well for the general case of possibly singular
distributions. \proofend

\xdef\hasexteprop{\the\sectno.\the\theono} 
\theo
$GEO({\cal M})$ has the extension property (cf. Definition \extenproper).

\beginproof  Suppose that $\xi\in\maxi{GEO({\cal M})}$, $I=I(\xi)$, and
$\tau$ is a limit point of $I$ such that the limit  $x=\lim_{t\to\tau,t\in
I}\xi(t)$ exists.  Let $\Xi\in\widehat{GEO}({\cal M})$ be such that
$\pi_{_{T^*M}}\compo\Xi=\xi$, and write $\Xi(t)=(\xi(t),\psi(t))$ for 
$t\in I$. We have to prove that $t$ is an interior point of $I$. Pick
a neighborhood $U$ of $x$ which is contained in the domain of a
coordinate chart $\bx$ and in that of an orthonormal basis of sections
${\bf f}=(f_1,\ldots,f_m)$ of $F$.  
Then the right-hand side of (\geosystem) is bounded along $\Xi$ 
for $t\in\,]\,\tau-\eps,\tau+\eps\,[\,\cap\, I$, for some $\eps>0$, because of
Gronwall's inequality and the fact that 
$\sum_{i=1}^m\mylangle\psi(t),g_i(\xi(t))\myrangle^2={\rm constant}$. 
Therefore $\lim_{t\to\tau,t\in I}\Xi(t)$ exists. So, by standard properties
of ordinary differential equations, $\Xi$ extends to a full neighborhood 
$\,]\,\tau-\eps,\tau+\eps\,[\,$ of $\tau$, and then 
$\xi$ extends to a geodesic defined on 
$\,]\,\tau-\eps,\tau+\eps\,[\,\cup\, I$. Since
$\xi\in\maxi{GEO({\cal{}M})}$,  we have 
$\,]\,\tau-\eps,\tau+\eps\,[\,\subseteq I$, so $\tau$ is an interior point
of $I$, as desired.\proofend

\section{Strongly bracket generating subriemannian structures}


\medbreak

\noindent If $x\in M$, we let
$E(x)^{\perp}\,\,\equaldef\,\,E^{\perp}(x)\,\,\equaldef\,\,\Big\{p\in T_x^*M: \mylangle p,v\myrangle=0\,\,\,{\rm for\,\,\,all}
\,\,\,v\in E(x)\Big\}$.
For each point $(x,p)\in T^*M$ such that $p\in E(x)^{\perp}$, we define a
bilinear form $Z(x,p)$ on $F(x)$ as follows. Let $v,w\in F(x)$, and let
$V$, $W$ be smooth sections of $F$ such that $V(x)=v$ and $W(x)=w$. Then
$\sigma(V)$ and $\sigma(W)$ are smooth vector fields on $M$, so the Lie
bracket $[\sigma(V),\sigma(W)]$ is well defined, and we can let
$Z(x,p)(v,w)=\Big\langle p,[\sigma(V),\sigma(W)]\Big\rangle$.  The number
$Z(x,p)(v,w)$ does not depend on the choice of the sections $V$, $W$.  (If,
for example, $V_1$ is another section of $F$ such that $V_1(x)=v$, then we
can choose a local basis ${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$ near
$x$, and write $V_1-V=\sum_i\zeta^if_i$ near $x$, where the $\zeta^i$ are
smooth functions that vanish at $x$. If $g_i=\sigma(f_i)$, then
$\sigma(V_1)-\sigma(V)=\sum_i\zeta^ig_i$, so
\xdef\svv{\equanosectno}
$$
[\sigma(V_1)-\sigma(V),\sigma(W)]=\sum_i\zeta^i[g_i,\sigma(W)]-
\sum_i(\sigma(W)\zeta^i).g_i\,.\numeq
$$
Since $\mylangle p,g_i(x)\myrangle=0$ ---because $p\in E(x)^{\perp}$---
and $\zeta^i(x)=0$, it follows from (\svv) that
$\mylangle p,[\sigma(V_1),\sigma(W)](x)\myrangle
=\mylangle p,[\sigma(V),\sigma(W)](x)\myrangle$, as desired.)

It is then clear that $Z(x,p)$ is always skew-symmetric.

\xdef\defofsbgdist{\the\sectno.\the\theono}
\defi
We call $E$ ---and ${\cal M}$--- {\it strongly bracket-generating} (SBG) at
a point $x\in M$ if the bilinear form $Z(x,p)$ is nonsingular for every 
$p\in E(x)^{\perp}\backslash\{0\}$. We call $E$ ---and ${\cal M}$--- {\it
strongly bracket-generating} if it is SBG at every point of $M$. \proofend

\rema
{\it A Riemannian manifold ---or, more generally, 
a subriemannian manifold for which
$\sigma$ is surjective, cf. Remark \remonriem{}--- 
is always SBG}, because in the Riemannian case
the condition of Definition \defofsbgdist{} holds vacuously, since 
$E(x)^{\perp}=\{0\}$. \proofend

The definition given above of the SBG condition is manifestly
intrinsic. We now give a local characterization of the SBG condition
in  terms of a choice of a basis of sections. 

Suppose ${\bf f}=(f_1,\ldots,f_m)$ is a local basis of smooth sections of
$F$. Write $g_i=\sigma(f_i)$. Define a vector field ${\bf{}g}\theta$, for
each	 $\theta=(\theta^1,\ldots,\theta^m)\in\rr^m$, by letting 
${\bf{}g}\theta=\theta^1g_1+\ldots+\theta^mg_m$.  We prove in the appendix
that 

\vbox{
\xdef\sbgoldchar{\the\sectno.\the\theono}
\prop
A singular distribution $E=(F,\sigma)$ is strongly bracket-generating at a
point $x$ in the domain $U_{\bf f}$ of a local basis of sections of $F$
if and  only if

{\parindent=25pt


\medbreak
\itemitem{(SBG$_{loc}$)}  {\it for every 
$\theta\!=\!(\theta^1,\ldots,\theta^m)\!\in\!\rr^m\backslash\{0\}$, the 
tangent space $T_xM$ is spanned by the $2m$
vectors $g_1(x)\,,\,\ldots\,,\,g_m(x)\,,\,[{\bf g}\theta,g_1](x)\,,\,\ldots
\,,\,[{\bf g}\theta,g_m](x)$.} \proofend

}

}

\xdef\sbgcoro{\the\sectno.\the\theono}
\coro
If a singular distribution $E=(F,\sigma)$ on a manifold $M$ is strongly
bracket-generating  at a point $x$ then $E$ is 2-generating at $x$. \proofend

The following elementary observation explains the importance of the SBG
condition:    

\xdef\noabnormal{\the\sectno.\the\theono}
\theo
If ${\cal M}$ is SBG, then a nonconstant extremal cannot be abnormal. 

\beginproof
We first prove the following ``coordinate free version of the adjoint
equation'' (cf.  Sussmann [\susscofree]):
\xdef\propbasicfla{\the\sectno.\the\theono}
\prop
Let ${\bf f}=(f_1,\ldots,f_m)$ be a local orthonormal basis of sections of
$F$, and let $g_i=\sigma(f_i)$.  Let $\xi:[a,b]\to U_{\bf f}$ be an
$E$-admissible $E$-arc.  Let $\eta$ be the
canonical control representation $\eta_{{\bf f},\xi,[a,b]}$ of $\xi$ 
on $[a,b]$ relative to
${\bf f}$, so $\eta(t)=(\eta^1(t),\ldots,\eta^m(t))$, and
$\sum_{i=1}^m\eta^i(t)f_i(\xi(t))=\omega_{\xi}(t)$.
Let $\psi$ be an absolutely continuous field of covectors along $\xi$ that
satisfies the adjoint equation (\adjequa).  Let $\Xi(t)=(\xi(t),\psi(t))$.
Then 
\xdef\basicflagen{\equanosectno}
$$
\frac{d}{dt}\Bigl\langle\psi(t),X(\xi(t))\Bigr\rangle=
\sum_{i=1}\eta^i(t).\Big\langle\psi(t),[g_i,X](\xi(t))\Big\rangle\,
\quad{\rm{}a.\,\,e.}\numeq$$ 
---that is, $\frac{d}{dt}\Big(\mu_X(\Xi(t))\Big)=
\sum_{i=1}\eta^i(t).\mu_{[g_i,X]}(\Xi(t))$---
\ for every smooth vector field $X$ on $M$. In particular, 
\xdef\basicfla{\equanosectno}
$$
\frac{d}{dt}\Big(\ffi_{{\bf f},j}(\Xi(t))\Big)=
\sum_{i=1}^m\eta^i(t).\zeta_{{\bf f},ij}(\Xi(t))
\,\,\,{\rm a.e.}\numeq 
$$
for $j=1,\ldots,m$, where 
\xdef\basicflazeta{\equanosectno}
$$
\zeta_{{\bf f},ij}(x,p)\,\,\equaldef\,\,\mylangle
p,[g_i,g_j](x)\myrangle\,.\numeq 
$$

\medbreak\noindent{\it Proof of Proposition \propbasicfla}.\ In 
coordinates, the adjoint equation says that (\basicflagen) holds when 
$X={\partial\,\,\,\,\over\partial x^i}$, $i=1,\ldots,$. An elementary
computation shows that the set of smooth vector fields  
$X$ for which (\basicflagen) holds is a $C^\infty(M)$-module. So
(\basicflagen) holds for all $X$. \proofend


\medbreak\noindent{\it Proof of Theorem \noabnormal}.\ Let 
$\xi$ be an abnormal extremal. We want to prove that $\xi(t)$ is
constant. It suffices to work locally, so we may assume that 
$\xi$ is contained in the domain $U_{{\bf f}}$ of an orthonormal
basis ${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$. Let $\psi$ be such
that the conditions of Definition \defofextremal{} hold with $\psi_0=0$,
and let $\Xi(t)=(\xi(t),\psi(t))$. 
Then the Hamiltonian maximization condition implies that the functions 
$t\to\ffi_{{\bf f},i}(\Xi(t))$ vanish identically, so 
the left-hand side of (\basicfla) vanishes identically. 
The fact that 
$\psi(t)\neq 0$ (which is a consequence of (E.1), since $\psi_0=0$)
implies then, since ${\cal M}$ is SBG, that the matrix 
$\Big(\zeta_{{\bf f},ij}(\Xi(t))\Big)_{ij}$ is nonsingular for each $t$. 
So  $\eta\equiv 0$, and then $\xi$ is a constant curve. \proofend

\xdef\psiunique{\the\sectno.\the\theono}
\theo
Assume ${\cal M}$ is SBF. Let $\xi$ be a nonconstant geodesic. Then the 
cotangent lift $\Xi\in\widehat{GEO}({\cal M})$ of $\xi$ is unique. 

\beginproof Clearly, we may work locally, so we will assume that $M=\rr^n$
and there exists a a global orthonormal basis ${\bf{}f}=(f_1,\ldots,f_m)$
of sections of $F$. Write $g_i=\sigma(f_i)$, and let $\Xi_j=(\xi,\psi_j(t))$,
$j=1,2$, be two distinct solutions of the geodesic equation.  Then
$\psi_1(t)\neq\psi_2(t)$ for every $t$ because if $\psi_1(t)=\psi_2(t)$ for
some $t$ then $\Xi_1\equiv\Xi_2$ by the uniqueness of solutions of ordinary
differential equations. 

Then, if $\ell\in\{1,2\}$,
$\dot\xi(t)=\sum_{i=1}^m\ffi_{{\bf f}}^i(\Xi_\ell(t))g_i(\xi(t))$.
Proposition \iscanonical{} 
then tells us that both vector functions $t\to\tilde\ffi_{{\bf f}}(\Xi_j(t))$ 
are equal to the canonical control representation $\eta_{{\bf f},\xi,I(\xi)}$. 
So  $\ffi_{{\bf f}}(\Xi_1(t))=\ffi_{{\bf f}}(\Xi_2(t))$ for a.~e. $t$. 
We then use Equation (\basicfla) and conclude that
$\frac{d}{dt}\Big(\ffi_{{\bf f},j}(\Xi_\ell(t))\Big)=
\sum_{i=1}^m\eta_{{\bf{}f},\xi}^i(t)\zeta_{{\bf f},ij}(\xi,\psi_\ell(t)))$
a.~e.~\ for $j=1,\ldots,m$, $\ell=1,2$.  So 
\xdef\buthten{\equanosectno}
$$
\sum_{i=1}^m\eta_{{\bf{}f},\xi}^i(t)
\Big(\zeta_{{\bf f},ij}(\xi(t),\psi_1(t)-\psi_2(t))\Big)=0\numeq
$$
for $j=1,\ldots,m$. For each $t$, $\psi_1(t)\neq \psi_2(t)$ but 
$\ffi_{{\bf f}}(\Xi_1(t))=\ffi_{{\bf f}}(\Xi_2(t))$. This implies that
$\psi_1(t)-\psi_2(t)\neq 0$ but 
$\ffi_{{\bf f}}(\xi,\psi_1(t)-\psi_2(t))=0$. It then follows from the
SBG condition that the matrix 
$\Big(\zeta_{{\bf f},ij}(\xi(t),\psi_1(t)-\psi_2(t))\Big)_{ij}$ is
nonsingular. But then (\buthten) implies that 
$\eta_{{\bf f},\xi,I(\xi)}(t)\equiv 0$.
So $\xi(t)\equiv{\rm constant}$. \proofend

\rema
The conclusion of Theorem \psiunique{} is obviously false if $\xi$ is a
constant curve, unless $\sigma$ is surjective.  Indeed, if $x\in M$ is such
that $\sigma_x$ is not surjective, then we can choose $I$ arbitrarily,
and let $\xi:I\to M$ be the constant curve such that $\xi(t)\equiv x$. 
Pick a $p$ such that $\mylangle p,\sigma_x(v)\myrangle=0$ for all $v\in F(x)$,
and define $\psi(t)\equiv p$. Then $t\to(\xi(t),\psi(t))$ is in
$\widehat{GEO}({\cal M})$, and it is clear that $p$ is not unique. \proofend

\xdef\mingeosame{\the\sectno.\the\theono}
\theo
Let ${\cal M}$ be SBG, and let $\xi$ be a curve in $M$. Then $\xi$ is a
local energy minimizer if and only if it is a geodesic curve.

\beginproof
The conclusion is evidently true if $I(\xi)$ is trivial, since in this case
$\xi$ is both a geodesic and an energy minimizer.

Now suppose that $I(\xi)$ is nontrivial. 
The fact that if $\xi$ is a geodesic curve then $\xi$ is a local energy
minimizer is a general property of subriemannian manifolds, as shown 
earlier in Theorem \geoislocmin{}.  To prove the converse, 
we assume that $\xi$ is a local energy minimizer.  This means 
that for every $t\in I$ there exists a compact subinterval 
$[a_t,b_t]$ of $I$ such that the restriction
$\xi_t\,\equaldef\,\xi\restr[a_t,b_t]$ is an energy minimizer, 
and $t$ belongs to the interior $J_t$ of $[a_t,b_t]$ relative to $I$. 
Then each $\xi_t$ is parametrized by constant times arc-length,
so for each $t$ there exists a unique $L_t\in\rr$ such that 
$||\dot\xi(t)||_G=L_t$ for a.~e. $t\in J_t$. (The uniqueness follows 
because $J_t$ is nontrivial, since $I$ is.) 
It then follows that, if $J_t\cap J_s\neq\emptyset$, then 
$L_t=L_s$. (Indeed, $J_t\cap J_s$ is relatively open in $I$, and nonempty,
so it is nontrivial.) Since $I$ is connected, all the $L_t$ are equal,
so there exists $L\in\rr$ such that 
$||\dot\xi(t)||_G=L$ for a.~e. $t\in I$. Therefore $\xi$ is parametrized by
constant times arc-length. 

Now, if $L=0$ then $\xi\equiv{\rm constant}$, so
$\xi=\pi_{_{T^*M}}\compo\Xi$, where $\Xi(t)=(\xi(t),0)$ for all $t\in
I(\xi)$. Clearly, $\Xi\in\widehat{GEO}({\cal M})$, so $\xi$ is a geodesic.

If $L\neq 0$ then every curve $\xi_t$ is nonconstant, so it is a
geodesic.  Moreover, the adjoint vector $J_t\ni s\to\psi_t(s)\in
T_{\xi(s)}^*M$ such that the curve $[a_t,b_t]\ni
s\to\Xi_t(s)=(\xi(s),\psi_t(s))$ is a geodesic is unique by Theorem
\psiunique{}.    If $J_t\cap J_s\neq\emptyset$, then
$\psi_t$ and $\psi_s$ must agree on $J_t\cap J_s\neq\emptyset$, using
Theorem \psiunique{} once again.  So there exists a 
map $I\ni t\to\psi(t)\in T_{\xi(t)}^*M$ such that 
$\psi(s)=\psi_t(s)$ whenever $s\in J_t$.  Then 
$\psi$ is locally absolutely continuous, and the curve
$I\ni t\to\Xi(t)=(\xi(t),\psi(t))$ is in $\widehat{GEO}({\cal M})$. So
$\xi$ is a geodesic. \proofend

\xdef\lminshaveext{\the\sectno.\the\theono}
\coro 
If ${\cal M}$ is SBG, then $LMin^{PAL}({\cal{}M})$ has the extension property.

\beginproof
Let $\xi\in\maxi{LMin^{PAL}({\cal M})}\,$. 
Let $\tau$ be a limit point of $I(\xi)$ such that 
the limit $x=\lim_{t\to\tau,t\in I}\xi(t)$ exists.
Then $\xi$ is a local energy minimizer, so it is a
geodesic by Theorem \mingeosame. If $\tilde\xi:\tilde I\to M$ was a
geodesic that extends $\xi$ to a strictly larger interval, then $\tilde\xi$
would also be a local energy minimizer, by Theorem \mingeosame{}\  (or by
Theorem \geoislocmin{}), contradicting the maximality of $\xi$ within
$LMin^{PAL}({\cal M})$. So $\tilde\xi$ cannot exist, and then 
$\xi\in\maxi{GEO({\cal M})}$.
By Theorem \hasexteprop{}, $GEO({\cal M})$ has the extension property. 
Since $\xi\in\maxi{GEO({\cal M})}$ and 
$\tau$ is a limit point of $I(\xi)$ such that
$\lim_{t\to\tau,t\in I}\xi(t)$ exists, it follows that $\tau$ is an
interior point of $I(\xi)$. \proofend

Thanks to Theorem \mingeosame{} and Corollary \lminshaveext{}, the
subriemannian Hopf-Rinow theorem takes a particularly form in the SBG case:

\xdef\hopfrinowsbg{\the\sectno.\the\theono}
\theo
If ${\cal M}$ is SBG then the following six conditions are equivalent: 

{\sl

{\parindent=25pt

\item{(i)\phantom{i}\phantom{i}} ${\cal M}$ is complete;

\item{(ii)\phantom{i}} every closed ball $B_{{\cal M}}(x,r)$ is
compact; 

\item{(iii)} if $\{x_j\}_{j=1}^\infty$ is any sequence of 
points of $M$ whose distances to a given point $\bar x$ are bounded above
by a finite constant, then 
there exist a subsequence of $\{x_j\}_{j=1}^\infty$ that converges to
a limit $x$, and a length minimizer $\xi$ such that
$\partial\xi=(\bar x, x)$;

\item{(iv)} every $\xi\in\maxi{{\cal C}^{a,\le 1}({\cal M})}$ 
satisfies $I(\xi)=\rr$;

\item{(v)} every  $\xi\in\maxi{LMin^{PAL}({\cal M})}$ satisfies $I(\xi)=\rr$; 

\item{(vi)} every $\xi\in\maxi{GEO({\cal M})}$ satisfies 
$I(\xi)=\rr$. \proofend

} 

}




\section{The main estimate}

\noindent We are now ready to prove our main adjoint vector estimate, 
and derive
some consequences. If $S\subseteq M$, we let ${\cal G}_{nc,min}(S)$ 
denote the set of all 
$\Xi\in C^0([0,1],\pi_{_{T^*M}}^{-1}(K))\cap\widehat{GEO}({\cal M})$
such that the projection $\xi=\pi_{_{T^*M}}\compo\Xi$ is a
nonconstant energy minimizer.

The following result is then our adjoint vector estimate:
\xdef\mainesti{\the\sectno.\the\theono}
\theo
Let ${\cal M}=(M,F,\sigma,G)$ be an SBG subriemannian manifold, and let 
$K$ be a compact subset of $M$. Then ${\cal G}_{nc,min}(K)$ is a relatively
compact subset of $C^0([0,1],T^*M)$.  

\beginproof  It suffices to show that 

\medbreak

\item{(A)} {\it for every compact subset $K$ of $M$ there exists a compact
subset $J_K$ of $T^*M$ such that 
${\cal G}_{nc,min}(K)\subseteq C^0([0,1],J_K)$.}  


\medbreak\noindent
Indeed, if (A) holds, then ${\cal G}_{nc,min}(K)$ is uniformly bounded.
In addition  
${\cal G}_{nc,min}(K)$ is equicontinuous, because all its members are integral
arcs of the vector field $\vec{H}_{{\cal M}}$. So 
${\cal G}_{nc,min}(K)$ is relatively compact in  $C^0([0,1],T^*M)$ by the
Ascoli-Arzel\`a theorem.  

We now prove (A). First of all, we show that the desired result is local: 

{\sl 

{\parindent=35pt


\medbreak
\item{(LOC)} if $\bar K$ is a compact subset of $M$ such that 
every point $x\in\bar K$ has a compact neighborhood $K_x$ for which
there exists a compact subset $J_x$ of $T^*M$ with the property
that ${\cal G}_{nc,min}(K_x)\subseteq C^0([0,1],J_x)$, then 
there exists a compact subset  $\bar J$ of $T^*M$ such
that ${\cal G}_{nc,min}(\bar K)\subseteq C^0([0,1],\bar J)$.

}

}

\medbreak\noindent
To prove (LOC), we pick a covering of  $\bar K$ by finitely many open sets
$U_1,\ldots,U_\nu$ whose closures $K_1,\ldots,K_{\nu}$ are compact and such
that there are compact subsets $J_i$ of $T^*M$  with the property that
${\cal G}_{nc,min}(K_i)\subseteq C^0([0,1],J_i)$. We then let $\delta>0$ be a
Lebesgue number for the covering $\{U_1,\ldots,U_\nu\}$ of $\bar K$
relative to the metric $d_{{\cal M}}$, i.e. a number such that every subset
of $\bar K$ of $d_{{\cal M}}$-diameter $\le\delta$
is contained in one of the $U_i$.   We let $\Lambda$ be the $d_{{\cal
M}}$-diameter of 
$\bar K$, and pick a positive integer $N$ such that $\Lambda\le N\delta$. 

Let $\Xi\in{\cal G}_{nc,min}(\bar K)$, and write
$\xi=\pi_{_{T^*M}}\compo\Xi$. Then 
$\xi$ is parametrized by constant times arc-length, 
so $||\dot\xi(t)||_{_G}=||\xi||_{_G}$ for all $t\in[0,1]$. Clearly,
$0<||\xi||_{_G}\le\Lambda$. So, if we let $I_k$ be the closed interval 
$\Big[{k-1\over N},{k\over N}\Big]$, for $k=1,\ldots,N$, and use 
$\xi_k$ to denote the restriction of $\xi$ to $I_k$, then each
$\xi_k$ is entirely contained in one of the sets $U_i$, since 
$d_{{\cal M}}(\xi(t),\xi(t'))\le\Lambda N^{-1}\le\delta$ whenever $t,t'\in I_k$. 
Let $\theta_k(t)=\xi\Big({k-1+t\over N}\Big)\quad{\rm for}\quad t\in[0,1]$.
Then $\theta_k$ is the canonical reparametrization of $\xi_k$.  So
\xdef\noncon{\equanosectno}
$$
||\dot\theta_k(t)||_{_G}={||\xi||_{_G}\over N}\quad{\rm for\,\,\,a.\,e.}\quad t\,.\numeq
$$
Let $\Xi(t)=(\xi(t),\psi(t))$ for $t\in[0,1]$. Then, if we define
$\psi_k(t)={1\over N}.\psi\Big({k-1+t\over N}\Big)$ for $t\in[0,1]$, 
an easy computation shows that each of the curves
$t\to(\theta_k(t),\psi_k(t))$ is in $\widehat{GEO}({\cal M})$. 
Moreover, $\theta_k$ is nonconstant, because (\noncon) implies that 
$||\dot\theta_k(t)||_{_G}\neq 0$ for a.~e. $t$.  So $(\theta_k(t),\psi_k(t))\in
J_k$ for $t\in[0,1]$, $k=1,\ldots,N$.  Let 
$\bar J=\bigcup_{k=1}^{\nu}\Bigl\{(x,p):\Big(x,{p\over N}\Big)\in
J_k\Bigr\}$. Then $\Xi$ is entirely contained in $\bar J$.   So (LOC) is
proved. 

We now assume that $M$ is in fact $\rr^n$, $K$ is the closed unit
ball $\{x\in\rr^n:||x||\le 1\}$, and $F$ has a global orthonormal basis 
${\bf f}=(f_1,\ldots,f_m)$ of sections.  From now on, we fix the 
local basis ${\bf f}$,
and write $g_i=\sigma(f_i)$. We write 
$\tilde\ffi^i$, $\ffi_i$, $\zeta_{ij}$, rather than 
$\tilde\ffi_{{\bf f}}^i$, $\ffi_{{\bf f},i}$, 
$\zeta_{{\bf f},ij}$, so 
$\ffi_i(x,p)=\mylangle p,g_i(x)\myrangle$ and 
$\zeta_{ij}(x,p)=\mylangle p,[g_i,g_j](x)\myrangle$.   Also, we use
$\Phi(x,p)$, $Z(x,p)$ to denote, respectively, the column vector
with components $\tilde\ffi^1(x,p),\ldots,\tilde\ffi^m(x,p)$ and the
$m\times m$ matrix $\Big(\zeta_{ij}(x,p)\Big)_{1\le i,j\le m}$. 

Now $T^*M$ is just the product $\rr^n\times\rr^n$.
For vectors $v\in\rr^n$, $||v||$ will always mean Euclidean norm. For matrices
$A$ in $\rr^{\mu\times \nu}$, $||A||$  will denote operator norm, 
corresponding to regarding $A$ as linear map from $\rr^\nu$ to $\rr^\mu$.
We let $\tilde K$ be the ball $\{x\in\rr^n:||x||\le 2\}$. Using 
Proposition \propestimate, we pick positive constants $\Delta_1$, 
$\Delta_2$,  such that 
\xdef\sqrootbound{\equanosectno}
$$
\Delta_1||x-x'||\le d_{{\cal M}}(x,x')\le\Delta_2||x-x'||^{1/2}
\quad{\rm for\,\,\,all}\quad x,x'\in \tilde K\,.\numeq
$$ 
Our goal is to prove that an {\it priori}
bound 
\xdef\whatwewant{\equanosectno}
$$
\kappa_+(\Xi)\,\,\equaldef\,\,\sup\Big\{||\psi(t)||:0\le 1\Big\}\le C\numeq
$$
holds for all $\Xi=(\xi,\psi)\in{\cal G}_{nc,min}(K)$, with a constant 
$C$ that does not depend on $\Xi$. 

So from now on we fix a $\Xi\in{\cal G}_{nc,min}(K)$, write 
$\Xi(t)=(\xi(t),\psi(t))$, and try to prove (\whatwewant), making sure that 
$C$ that does not depend on $\Xi$. We let $\eta:[0,1]\to\rr^m$ be the
canonical control representation $\eta_{{\bf f},\xi,[0,1]}$ 
of $\xi$ with respect to ${\bf f}$.
We let $L(\xi)=||\xi||_{_G}$, and write $\partial\xi=(\bar x,\hat x)$, so
$\bar x=\xi(0)$, $\hat x=\xi(1)$, and $d_{{\cal M}}(\bar x,\hat x)=L(\xi)$.

We let $\Lambda$ denote the diameter of $K$ relative to the
metric $d_{{\cal M}}$. We then know that $\eta(t)=\Phi(\Xi(t))$, 
i.~e. that $\eta^i(t)=\tilde\ffi^i(\Xi(t))$ for $i=1,\ldots,m$, and that 
$||\eta(t)||$ is a.~e. constant and equal to $L(\xi)$, so
$||\eta(t)||\le\Lambda$ for a.~e. $t$. Then 
\xdef\firstbound{\equanosectno}
$$
\Big|\mylangle\psi(t),g_i(\xi(t)\myrangle\Big|\!=\!|\ffi_i(\Xi(t))|\!=\!|\eta^i(t)|
\!\le\! L(\xi)\!\le\!\Lambda\,\,\,{\rm for\,\,\,all}\,\,\,t\!\in\![0,1]\,,
\,i\!\in\!\{1,\ldots,m\}\,. \numeq
$$
This gives us an {\it a priori} bound on the components of $\psi(t)$ 
in the directions of the vector fields $g_i$. To get (\whatwewant), we need
---since ${\cal M}$ is 2-generating (cf. Corollary \sbgcoro)---
an {\it a priori} bound on the components of $\psi(t)$ in the direction 
of the brackets $[g_i,g_j]$, i.~e. on the entries $\zeta_{ij}(\Xi(t))$ of
the matrix $Z(\Xi(t))$. 

The adjoint equation (\adjequa), together with the bound (\firstbound),
implies that 
\xdef\whatwewantca{\equanosectno}
$$
||\dot\psi(t)||\le\Lambda.\Theta_1.||\psi(t)||\,,\quad{\rm where}\quad
\Theta_1\equaldef\sup\Biggl\{\sum_{i=1}^m\Big\|{\partial g_i\over\partial
x\,\,}(x)\Big\|:x\in K\Biggr\}\,.\numeq
$$
So $||\psi(t)||\le e^{\Lambda.\Theta_1}||\psi(t')||$ whenever
$t,t'\in[0,1]$. It follows that
\xdef\whatwewantc{\equanosectno}
$$
\kappa_+(\Xi)\le e^{\Lambda.\Theta_1}.\kappa_-(\Xi)\,,\quad{\rm where}\quad
\kappa_-(\Xi)\,\,\equaldef\,\,\min\Big\{||\psi(t)||:0\le 1\Big\}\,.\numeq
$$
Therefore, it suffices to derive an {\it a priori} bound for
$\kappa_-(\Xi)$.

Formula (\basicfla) says that
\xdef\basicflavec{\equanosectno}
$$
\frac{d}{dt}\Biggl(\Phi(\Xi(t))\Biggr)=-Z(\Xi(t)).\Phi(\Xi(t))\quad{\rm
for\quad a.e.}\quad t\,.\numeq 
$$
We now write 
\xdef\weuse{\equanosectno}
$$
\hat x-\bar
x=\xi(1)-\xi(0)=\int_0^1{\bf{}g}(\xi(t)).\Phi(\Xi(t))\,dt\,\,\equaldef
I\numeq 
$$
where, if $x\in\rr^n$, ${\bf{}g}(x)$ is the $n\times m$ matrix 
whose columns are the vectors $g_1(x),\ldots,g_m(x)$, so 
that, if $u=(u^1,\ldots,u^m)$ is any column $m$-vector, then 
${\bf{}g}(x).u$ stands for $\sum_{i=1}^mu^ig_i(x)$. 

It follows from (\sqrootbound) that 
\xdef\weuseb{\equanosectno}
$$
L(\xi)^2=d_{{\cal M}}(\bar x,\hat x)^2\le\Delta_2^2||\hat x-\bar x||\,.\numeq
$$
We would like to use (\weuse) to prove a bound of the form 
$||\hat x-\bar x||\le {\rm constant}.L(\xi)^2\kappa_-(\xi)^{-1}$. Since 
$L(\xi)\neq 0$, this bound, together with (\weuseb), would imply an upper
bound for $\kappa_-(\xi)$.  

First of all, it is clear that
\xdef\weusexxx{\equanosectno}
$$
||\hat x-\bar x||\le\Theta_2.L(\xi)\,,\quad{\rm where}\quad
\Theta_2=\sup\Biggl\{\sum_{i=1}^m||g_i(x)||:x\in\tilde K\Biggr\}\,.\numeq
$$
This is not yet the $L(\xi)^2$ estimate we are seeking, but will be useful
later. 

In order to get a better bound, we use integration by parts to rewrite
the integral of (\weuse).  The idea is to write
$\Phi=Z^{-1}.Z.\Phi=-Z^{-1}.\dot\Phi$, so that the integral will become
$\int_0^1{d\,\,\over dt}\Bigl(Z(\Xi(t))^{-1}\Bigr)\Phi(\Xi(t))\,dt+{\rm
boundary\,\,\,terms}$. The derivative of 
$Z(\Xi(t))^{-1}$ will involve the control ---i.~e. $\Phi(\Xi(t))$---
again, and this will give an $L(\xi)^2$-bound for 
$||\hat x-\bar x||$. 

This, however, does not work directly as explained, because $Z(\Xi(t))$
may fail to be invertible. So we need a better argument, for which 
some preliminary work is required. 

We let $S=\rr^n\times(\rr^n\backslash\{0\})$, and use 
$\Sigma_k$, for any integer $k$, to denote
the set of all  smooth real-valued functions $h$ on $S$ that are
positively homogeneous of degree $k$ with respect to $p$, i.~e. such that 
$h(x,rp)=r^kh(x,p)$ whenever $(x,p)\in S$ and $r>0$.  

In order to prove our bound, we will assume, to begin with, that 
\xdef\assua{\equanosectno}
$$
\kappa_-(\Xi)\ge 1\,.\numeq
$$
This implies, in particular, that 
$\Xi$ is entirely contained in $S$. 

We let $D(x,p)$ denote the determinant of the matrix
$Z(x,p)$, so $D\in \Sigma_{m}$. We then define 
$A_1(x,p)=||p||^{-2m}\Big(||\Phi(x,p)||^{2m}+D(x,p)^2\Big)$
for $p\neq 0$. 
Then $A_1\in\Sigma_{0}$. The SBG condition implies that $A_1(x,p)$ never
vanishes. So we can define  $A_2(x,p)=A_1(x,p)^{-1}$, 
and then $A_2\in\Sigma_{0}$ as well.   We then rewrite the integral
of (\weuse) as
$$
\int_0^1{\bf{}g}(\xi(t)).\Phi(\Xi(t))\,dt=I_1+I_2\,,
$$
where 
$$
I_1=\int_0^1{||\Phi(\Xi(t))||^{2m}\over||\psi(t)||^{2m}}.A_2(\Xi(t)).
{\bf{}g}(\xi(t)).\Phi(\Xi(t))\,dt\,,
$$
and
$$
\!\!\!\!\!\!I_2=\int_0^1{D(\Xi(t))^2\over||\psi(t)||^{2m}}.A_2(\Xi(t)).
{\bf{}g}(\xi(t)).\Phi(\Xi(t))\,dt\,.
$$
Since $A_2\in\Sigma_0$, there is a constant $\Theta_3$ such that
the bound $|A_2(x,p)|\le\Theta_3$ holds
for all $(x,p)\in K\times(\rr^n\backslash\{0\})$.  Then 
\xdef\boundforione{\equanosectno}
$$
||I_1||\le\Theta_2\Theta_3 {L(\xi)^{2m+1}\over \kappa_-(\Xi)^{2m}}
\le\Theta_2\Theta_3\Lambda^{2m-1}{L(\xi)^{2}\over \kappa_-(\Xi)}\,,
$$
where we have used the facts that $L(\xi)\le\Lambda$, $m\ge 1$, and
$\kappa_-(\Xi)\ge 1$. 

We now need to estimate $I_2$. For this purpose, we let 
$\tilde Z(x,p)$ denote the matrix of complementary minors of $Z(x,p)$, so 
$\tilde Z(x,p)$ is an $m\times m$ matrix whose components belong to 
$\Sigma_{m-1}$, and $\tilde Z(x,p).Z(x,p)=D(x,p).{\bf 1}_m$, where 
${\bf 1}_m$ is the $m\times m$ identity matrix. Then
$$
\eqalign{
I_2&=\int_0^1{D(\Xi(t))\over||\psi(t)||^{2m}}.A_2(\Xi(t)).
{\bf{}g}(\xi(t)).\Big(D(\Xi(t)){\bf 1}_m\Big).\Phi(\Xi(t))\,dt\cr
&=\int_0^1{D(\Xi(t))\over||\psi(t)||^{2m}}.A_2(\Xi(t)).
{\bf{}g}(\xi(t)).\tilde Z(\Xi(t)).Z(\Xi(t)).\Phi(\Xi(t))\,dt\cr
&=-\int_0^1{\bf{}B}(\Xi(t)).{d\,\,\over dt}
\Big(\Phi(\Xi(t))\Big)\,dt\,,\cr}
$$
where ${\bf B}$ is an $n\times m$ matrix of smooth functions on $S$, given by 
$$
{\bf B}(x,p)=D(x,p).||p||^{-2m}.A_2(x,p).{\bf{}g}(x).\tilde Z(x,p)\,.
$$
It is then clear that the components of ${\bf B}$ are in 
$\Sigma_{-1}$. 

Integration by parts then yields $I_2=I_3+\bar\beta-\hat\beta$, where 
$$
I_3=\int_0^1{d\,\,\over dt}\Big({\bf{}B}(\Xi(t))\Big).
\Phi(\Xi(t))\,dt\,,\quad
\bar\beta={\bf{}B}(\Xi(0)).\Phi(\Xi(0))\,,\quad{\rm and}\quad
\hat\beta={\bf{}B}(\Xi(1)).\Phi(\Xi(1))\,.
$$
The derivative ${d\,\,\over dt}\Big({\bf{}B}(\Xi(t))\Big)$ is equal
to $\vec{H}_{{\cal M}}{\bf{}B}(\Xi(t))$, i.~e.
to $\sum_{i=1}^m\tilde\ffi^i(\Xi(t)).(G_i{\bf B})(\Xi(t))$, where 
$G_i$ is the Hamilton vector field $\vec{\mu}_{g_i}$ arising from
the momentum function $\mu_{g_i}$. Since $\mu_{g_i}\in \Sigma_{1}$,
the differential operators $G_i$ map $\Sigma_k$ into $\Sigma_k$ for every
$k$. So the components of the matrices $(G_i{\bf B})(x,p)$ 
are in $\Sigma_1$. Therefore there is a constant $\Theta_4$, depending only
on $K$,  such that
$||\vec{H}_{{\cal
M}}{\bf{}B}(\Xi(t))||\le\Theta_4.L(\xi).\kappa_-(\Xi)^{-1}$, and then 
\xdef\boundforithree{\equanosectno}
$$
||I_3||\le\Theta_4.{L(\xi)^2\over\kappa_-(\Xi)}\,.\numeq
$$
We have therefore shown that 
\xdef\idenxz{\equanosectno}
$$
\hat x-\bar x=I_1+I_3+\bar\beta-\hat\beta\,,\numeq
$$ 
where $I_1+I_3$ satisfies the bound 
\xdef\boundi1i3{\equanosectno}
$$
||I_1+I_3||\le\Theta_5.{L(\xi)^2\over\kappa_-(\Xi)}\,,\quad
\quad\Theta_5=\Theta_4+\Theta_2\Theta_3\Lambda^{2m-1}\,.\numeq
$$

We now have to find estimates for the boundary terms 
$\bar\beta$, $\hat\beta$ of the integration by parts.
Let ${\bf B}(x,p)=(b_1(x,p),\ldots,b_m(x,p))$. Then the vectors 
$b_i(x,p)$ are linear combinations $\sum_{j=1}^m\rho_i^j(x,p)g_j(x)$,
with coefficient functions $\rho_i^j\in\Sigma_{-1}$.
Then 
$$
\bar\beta=\sum_{i=1}^m\sum_{j=1}^m
\rho_i^j(\Xi(0)).\tilde\ffi^i(\Xi(0)).g_j(\bar x)=
\sum_{j=1}^m\bar v^j.g_j(\bar x)\,,
$$
where $\bar v^j=\sum_{i=1}^m\rho_i^j(\Xi(0))\tilde\ffi^i(\Xi(0))$, so that 
the vector $\bar v=(\bar v^1,\ldots,\bar v^m)$ satisfies
\xdef\bfbv{\equanosectno}
$$
||\bar v||\le{\Theta_6.L(\xi)\over\kappa_-(\Xi)}\,,\numeq
$$
for a constant $\Theta_6$ that does not depend on $\Xi$.  
This, of course, gives the bound
\xdef\boundbbeta{\equanosectno}
$$
||\bar\beta||\le{\Theta_2.\Theta_6.L(\xi)\over\kappa_-(\Xi)}\,,\numeq
$$
which is not yet sufficient for our purposes, but will be used later. 

We now let \ $t\to\gamma(t)$, \ $0\le t\le 1$, be the solution of the 
initial value problem
$\dot x=\sum_{i=1}^m\bar v^ig_i(x)$, $x(0)=\bar x$, 
and write $\bar y=\gamma(1)$, so $\bar y$ is the point of $\rr^n$ that is
reached from $\bar x$  in time $1$ 
following a trajectory of the control system $\dot x=\sum_{i=1}^mu^ig_i(x)$
with the constant control $\bar v$. Then 
$||\dot\gamma(t)||_{_G}\le||\bar{v}||$ for a.~e. $t$, so 
$||\gamma||_{_G}\le||\bar{v}||$, and then (\bfbv) yields
$d_{{\cal M}}(\bar y,\bar x)\le{\Theta_6.L(\xi)\over\kappa_-(\Xi)}\,.
$
So, if we assume that 
\xdef\assub{\equanosectno}
$$
\kappa_-(\Xi)\ge 6.\Theta_6\,,\numeq
$$
we find that 
\xdef\wefindt{\equanosectno}
$$
d_{{\cal M}}(\bar y,\bar x)\le{L(\xi)\over 6}\,.\numeq
$$
Next, we observe that the vector $\dot\gamma(t)$ satisfies 
$||\dot\gamma(t)||\le \Theta_2.\Theta_6.L(\xi).\kappa_-(\Xi)^{-1}$
as long as $\gamma(t)\in\tilde K$.  Let  us assume that 
\xdef\assuc{\equanosectno}
$$
\kappa_-(\Xi)\ge \Theta_2.\Theta_6.\Lambda\,.\numeq
$$
Then $||\dot\gamma(t)||\le 1$ as long as 
$||\gamma(t)||\le 2$. Since $\gamma(0)=\bar x\in K$, we have 
$||\gamma(0)||\le 1$, and then $\gamma(t)\in \tilde K$ for all $t\in[0,1]$.

We now compare the points $\bar z=\bar x+\bar\beta$ and 
$\bar y$.  We let $\gamma^{\#}(t)=\bar x+t\bar\beta$. 
Also, we let $(x,x')\to G_j(x,x')$ be smooth matrix-valued
functions with the property that  
\hbox{$g_j(x)-g_j(x')=G_j(x,x').(x-x')$ for $x,x'\in\rr^n$.} 
(For example, if $Dg_j$ is the Jacobian matrix of $g_j$, then 
we can take \ $G(x,x')=\int_0^1Dg_j(x+s(x'-x))\,ds$.)
\ Then, if we define
$B_1(t)=\sum_{j=1}^m\bar{v}^jG_j(\gamma(t),\gamma^{\#}(t))$ and 
$B_2(t)=\sum_{j=1}^m\bar{v}^jG_j(\gamma^{\#}(t),\bar x)$, we have 
$$
\eqalign{
\gamma(t)-\gamma^{\#}(t)&=\int_0^t\sum_{i=1}^m\bar v^j\Big(g_j(\gamma(t))-
g_j(\bar x)\Big)\,dt\cr
&=\int_0^t\sum_{i=1}^m\bar{v}^j\Big(g_j(\gamma(t))-
g_j(\gamma^{\#}(t))\Big)dt
+\int_0^t\sum_{i=1}^m\bar{v}^j\Big(g_j(\gamma^{\#}(t))\,dt
-g_j(\bar x)\Big)dt\cr
&=\int_0^tB_1(t).\Big(\gamma(t)\!-\!\gamma^{\#}(t)\Big)dt
+\int_0^t B_2(t).\Big(\!\gamma^{\#}(t)\!-\!\bar x\Big)dt\,\,.\cr}
$$
The vector $\gamma^{\#}(t)-\bar x$ is equal to 
$t\bar\beta$, which is bounded in norm by 
$\Theta_2.\Theta_6.L(\xi).\kappa_-(\Xi)^{-1}$ thanks to (\boundbbeta). So,
if  we let $\Theta_7=\sup\Biggl\{\sum_{j=1}^m||G_j(x,x')||:x,x'\in\tilde
K\Biggr\}$, then 
$$
\Bigl\|\int_0^tB_2(t).\Big(\gamma^{\#}(t)-\bar x\Big)\,dt\Bigr\|\le
\Theta_2.\Theta_6.\Theta_7.L(\xi)^2.\kappa_-(\Xi)^{-1}\,.
$$
Therefore Gronwall's inequality yields 
$$
||\gamma(t)-\gamma^{\#}(t)||\le\Theta_8{L(\xi)^2\over\kappa_-(\Xi)}\,,
\quad{\rm where}\quad 
\Theta_8=e^{\Lambda.\Theta_7}\Theta_2.\Theta_6.\Theta_7\,.
$$
In particular, 
\xdef\sotells{\equanosectno}
$$
||\bar y-\bar z||\le\Theta_8{L(\xi)^2\over\kappa_-(\Xi)}\,.\numeq
$$
On the other hand, (\sqrootbound) implies that 
$d_{{\cal M}}(\bar y,\bar z)\le\Delta_2||\bar y-\bar z||^{1/2}$. 
So (\sotells) tells us that 
\xdef\sotellsb{\equanosectno}
$$
d_{{\cal M}}(\bar y,\bar z)\le\Delta_2\Theta_8^{1/2}\kappa_-(\Xi)^{-1/2}
L(\xi)\,.\numeq
$$
If we now assume that 
\xdef\assud{\equanosectno}
$$
\kappa_-(\Xi)\ge 36\Delta_2^2\Theta_8\,,\numeq
$$
then (\sotellsb) becomes
$d_{{\cal M}}(\bar y,\bar z)\le{L(\xi)\over 6}$. 
Combining this and (\wefindt), we get
$d_{{\cal M}}(\bar z,\bar x)\le{L(\xi)\over 3}$. 
A similar computation yields
$d_{{\cal M}}(\hat z,\hat x)\le{L(\xi)\over 3}$, 
where $\hat z=\hat x+\hat\beta$.  Then 
$$
L(\xi)=d_{{\cal M}}(\bar x,\hat x)
\le d_{{\cal M}}(\bar x,\bar z)+d_{{\cal M}}(\bar z,\hat z)+
d_{{\cal M}}(\hat z,\hat x)\le {2L(\xi)\over 3}+
d_{{\cal M}}(\bar z,\hat z)\,,
$$
so 
\xdef\combi{\equanosectno}
$$
d_{{\cal M}}(\bar z,\hat z)\ge {L(\xi)\over 3}\,.\numeq
$$
On the other hand, (\idenxz) says that
\xdef\zhatzbar{\equanosectno}
$$
\hat z-\bar z=I_1+I_3\,.\numeq
$$
Then (\boundi1i3)  implies that 
\xdef\boundi1i3i{\equanosectno}
$$
||\hat z-\bar z||\le\Theta_5.{L(\xi)^2\over\kappa_-(\Xi)}\,.\numeq
$$
We now use (\sqrootbound) to conclude that
\xdef\combib{\equanosectno}
$$
d_{{\cal M}}(\hat z,\bar z)\le\Delta_2||\hat z-\bar z||^{1/2}
\le\Delta_2\Theta_5^{1/2}.L(\xi).\kappa_-(\Xi)^{-1/2}\,.\numeq
$$
If we combine (\combi) and (\combib), we get 
${L(\xi)\over 3}\le \Delta_2\Theta_5^{1/2}.L(\xi).\kappa_-(\Xi)^{-1/2}$.
Since $L(\xi)\neq 0$, this implies the inequality
\xdef\finbound{\equanosectno}
$$
\kappa_-(\Xi)\le 9\Delta_2^2\Theta_5\,.\numeq
$$
The bound (\finbound) has been established under the assumptions
that $\Xi\in{\cal G}_{nc,min}(K)$ and 
(\assua), (\assub), (\assuc), (\assud) hold.  It then follows that the 
bound $\kappa_-(\Xi)\le C$
holds for all $\Xi\in{\cal G}_{nc,min}(K)$, if we let 
$$
C=\max\Bigl(1\,\,,\,\,6.\Theta_6\,\,,\,\,\Theta_2.\Theta_6.\Lambda
\,\,,\,\,36\Delta_2^2\Theta_8\,\,,\,\,9\Delta_2^2\Theta_5\Bigr)\,.\numeq
$$
Therefore $\kappa_+(\Xi)\le C.e^{\Lambda.\Theta_1}$. 
So, if we let $J=\{(x,p):x\in K\,,\,||p||
\le C.e^{\Lambda.\Theta_1}\}$, we see that 
every $\Xi\in{\cal G}_{nc,min}(K)$ is  entirely contained in $J$. \proofend


\medbreak So far, we have studied the sets ${\cal G}_{nc,min}(K)$, whose
members $\Xi$ are such that $\pi_{_{T^*M}}\compo\Xi$ is entirely contained
in $K$. We would now like to discuss sets $\Sigma$ of arcs 
$\Xi\in C^0([0,1],T^*M)$ defined by the conditions that 
(a)~$\xi=\pi_{_{T^*M}}\compo\Xi$ is a  nonconstant energy minimizer,
and (b)~the endpoint condition $\partial\xi$ belongs to a given compact 
subset of $M\times M$. In the next section, these sets will
play a crucial role in the proof of the subanalyticity of the distance
function. 

It is now no longer  true in general that all our
arcs will be contained in a compact subset of $T^*M$.  On the other hand,
Theorem \mainesti{} trivially implies that

\xdef\trivially{\the\sectno.\the\theono}
\prop 
If $\Sigma$ is a subset of ${\cal G}_{nc,min}(M)$ with the property
that all the
projections $\xi=\pi_{_{T^*M}}\compo\Xi$, $\Xi\in\Sigma$, are contained in
a fixed compact subset of $M$, then $\Sigma$ is relatively compact in
$C^0([0,1],T^*M)$. \proofend

So it suffices to investigate the relative compactness
of the set of projections. 

If $\bar S,\hat S$ are subsets of $M$, we use 
${\cal EM}(\bar S,\hat S)$ to denote the set of all energy minimizers
$\xi:[0,1]\to M$ such that $\partial\xi\in\bar S\times\hat S$.
We would like to know whether ${\cal EM}(\bar K,\hat K)$ is
a compact subset of $C^0([0,1],M)$ if $\bar K$ and $\hat K$ are compact.


The answer is clearly ``no,'' even in the Riemannian case, unless some
extra conditions are imposed, as can be seen by taking $M$ to be the plane
$\rr^2$ minus one point $P$, and choosing $\bar K\subseteq M$ and $\hat
K\subseteq M$ to be convex, but such that the convex hull of $\bar K\cup
\hat K$ contains $P$. This suggests that the result may be true if one 
requires that whenever $\bar x\in \bar K$ and $\hat x\in \hat K$ it follows
that there is an energy minimizer $\xi:[0,1]\to M$ such that
$\partial\xi=(\bar x,\hat x)$.  Remark \the\sectno.4 below shows that this
condition is not yet sufficient, even for Riemannian metrics.

It turns out that, for SBG metrics, a slightly stronger hypothesis
suffices, as we now show.  Moreover, Theorem \mainesti{} turns out to play
a crucial role in the proof.

\xdef\bears{\the\sectno.\the\theono}
\theo
Assume that ${\cal M}$ is SBG. Let $\bar K,\hat K$ be
compact subsets of $M$ such that

{\sl

\item{($\#$)} there exists a neighborhood $U$ of $\bar K$ in $M$ with 
the property that, whenever $\bar x\in U$ and $\hat x\in\hat K$, it follows
that there exists a length minimizer $\xi$ such that 
$\partial\xi=(\bar x,\hat x)$. 

\noindent Then 

\item{(A)} There exists a compact subset ${\cal K}$ of 
$C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$ such that 
\xdef\compapropem{\equanosectno}
$$
{\cal EM}(\bar K,\hat K)=\Bigl\{\pi_{_{T^*M}}\compo\Xi:\Xi\in{\cal
K}\Bigr\}\,.\numeq
$$

\item{(B)} ${\cal EM}(\bar K,\hat K)$ is a compact subset of
$C^0([0,1],M)$. 

\item{(C)} The set $\Big\{\Xi\in{\cal G}_{nc,min}(M):
\pi_{_{T^*M}}\compo\Xi\in{\cal EM}(\bar K,\hat K)\Big\}$ is relatively
compact in the space $C^0([0,1],T^*M)$. 

}

\beginproof
It is clear that (B) is a trivial corollary of (A), since the map 
$$
C^0([0,1],T^*M)\ni\Xi\to\pi_{_{T^*M}}\compo\Xi\in C^0([0,1],M)
$$ 
is continuous. Also, Proposition \trivially{} implies that 
(C) follows from (B). So it suffices to prove (A). 

Clearly, (A) will follow if we prove: 

{\sl 

\medbreak
\itemitem{(A')} There exists a compact subset $J$ of $T^*M$ with the
property that, if $\xi:[0,1]\to M$ is any nonconstant energy minimizer such
that $\xi(0)\in\bar K$ and $\xi(1)\in\hat K$, and
$\Xi\in\widehat{GEO}({\cal{}M})$ is such that $\pi_{_{T^*M}}\compo\Xi=\xi$,
then $\Xi$ is entirely contained in $J$.

}

\medbreak\noindent Indeed, suppose (A') holds. Let ${\cal K}_0$ be the set
of all $\Xi\in C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$ 
such that $\Xi$ is contained in $J$ and $\pi_{_{T^*M}}\compo\Xi\in{\cal
EM}(\bar K,\hat K)$.  Then ${\cal K}_0$ is 
compact. ({\it Proof}$\,$: let $\{\Xi_j\}_{j=1}^\infty$ be a sequence in ${\cal
K}_0$. We want a subsequence that converges to a member of ${\cal
K}_0$. Since $C^0([0,1],J)\cap\widehat{GEO}({\cal M})$ 
is compact, it suffices to assume that the $\Xi_j$ converge in
$C^0([0,1],T^*M)$ to an arc $\Xi\in C^0([0,1],J)\cap\widehat{GEO}({\cal M})$.
If $\xi_j=\pi_{_{T^*M}}\compo\Xi_j$, and
$\xi=\pi_{_{T^*M}}\compo\Xi$, then $\partial\xi\in\bar K\times\hat K$,
because $\partial\xi_j\in\bar K\times\hat K$. Moreover, $\xi$ is a
canonically parametrized minimizer, because of Proposition
\isclosedprop. So $\Xi\in{\cal K}_0$.)  Let 
${\cal K}_1$ be the set of all constant curves $\Xi_{x}^{cons}:[0,1]\to
T^*M$, for all $x\in\bar K\cap\hat K$, where 
$\Xi_x^{cons}(t)\equaldef (x,0)$ for $t\in[0,1]$. 
Then ${\cal K}_1$ is obviously compact. 
Let ${\cal K}={\cal K}_0\cup{\cal K}_1$. Then ${\cal K}$ is 
compact, and (\compapropem) holds. 

So it suffices to prove (A').
We take $J$ to be the closure of the set $J_0$ of all
points  that are of the form $\Xi(t)$ for some 
$\Xi\in C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$  such that
$\xi=\pi_{_{T^*M}}\compo\Xi$ is a nonconstant length minimizer
and $\partial\xi\in\bar K\times \hat K$. We have 
to prove that $J$ is compact.  It suffices to show that 
every sequence $\{(x_j,p_j)\}$ of points of $J_0$ has a convergent
subsequence. So we pick one such sequence, and find curves 
$\Xi_j\in C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$  
such that $\xi_j=\pi_{_{T^*M}}\compo\Xi_j$ is a nonconstant length minimizer
and $\partial\xi_j\in\bar K\times \hat K$, and times $\tau_j\in[0,1]$
such that $(x_j,p_j)=\Xi_j(\tau_j)$. Write $\partial\xi_j=(\bar x_j,\hat x_j)$.
Then all the $\bar x_j$ are in $\bar K$, and all the $\hat x_j$ are in
$\hat K$.  We let $L_j$ denote the length of $\xi_j$. Then 
$L_j=d_{{\cal M}}(\bar x_j,\hat x_j)$, because $\xi_j$ is a
length minimizer. So $L_j\le\Lambda$, where 
$\Lambda=\sup\{d_{{\cal M}}(x,x'):x\in\bar K,x'\in\hat K\}$.  (It is clear
that $\Lambda<\infty$, because $d_{{\cal M}}$ is bounded on the compact set
$\bar K\times\hat K$.) Therefore, 
by passing to a subsequence, we may assume that the limits
$\bar x=\lim_{j\to\infty}\bar x_j$, $\hat x=\lim_{j\to\infty}\hat x_j$, 
$L=\lim_{j\to\infty}L_j$, and $\tau=\lim_{j\to\infty}\tau_j$
exist. Clearly, then,   $\bar x\in\bar K$, $\hat x\in\hat K$, $0\le\tau\le
1$, and $0\le L\le\Lambda$. 

Pick $\delta>0$ such that the closed ball 
$B=\{x\in M:d_{{\cal M}}(x,\bar x)\le 2\delta\}$
is compact and contained in $U$. Then pick $\alpha$ such that
$0<\alpha<1$ and $\alpha.\Lambda\le\delta$. By passing to a subsequence, we
may assume that $d_{{\cal M}}(\bar x_j,\bar x)\le\delta$ for all $j$. 
Then, since $\xi_j$ is parametrized by constant times arc-length, 
and $||\xi_j||_{_G}=L_j$, we have 
$||\dot\xi_j(t)||_{_G}= L_j$ for a.~e. $t$, so 
$d_{{\cal M}}(\xi_j(t),\bar x_j)\le tL_j\le t\Lambda$
for all $j$ and all $t$. Therefore 
$d_{{\cal M}}(\xi_j(t),\bar x_j)\le\delta$ for all $j$, if 
$t\in[0,\alpha]$. It follows that 
$\xi_j(t)\in B$ for all $j$ and all $t\in[0,\alpha]$. Moreover,
\xdef\neighb{\equanosectno}
$$
d_{{\cal M}}(\xi_j(t),\xi_j(t'))\le L_j|t-t'|\le \Lambda|t-t'|
\quad{\rm whenever}
\quad t,t'\in[0,\alpha]\,.\numeq
$$
So, if we let 
$\zeta_j$ denote the restriction of $\xi_j$ to the interval
$[0,\alpha]$, we see that $\{\zeta_j\}$ is an equicontinuous 
sequence of curves in the compact metric space $(B,d_{{\cal M}})$. 
Therefore, by passing to a subsequence we may assume that 
the $\zeta_j$ converge uniformly on $[0,\alpha]$ to a curve 
$\zeta:[0,\alpha]\to B$. It then follows from (\neighb) that 
\xdef\neighbb{\equanosectno}
$$
d_{{\cal M}}(\zeta(t),\zeta(t'))\le L|t-t'|\quad{\rm whenever}
\quad t,t'\in[0,\alpha]\,.\numeq
$$
Then Proposition \proplipimplies{} implies that $\zeta$ is an
$E$-admissible $E$-arc, and $||\dot\zeta(t)||_{_G}\le L$ for almost  all
$t\in[0,\alpha]$. Therefore $||\zeta||_{_G}\le L\alpha$.

On the other hand, since each $\zeta_j$ is a
length minimizer, the distance $d_{{\cal M}}(\bar x_j,\zeta_j(\alpha))$ 
is exactly equal to $L_j\alpha$. So $d_{{\cal M}}(\bar
x,\zeta(\alpha))=L\alpha$, because $d_{{\cal M}}$ is continuous. It follows
that $||\zeta||_{_G}=L\alpha$, and $\zeta$ is a length minimizer. 
Since $||\dot\zeta(t)||_{_G}\le L$ for almost  all $t\in[0,\alpha]$, we must
in fact have $||\dot\zeta(t)||_{_G}=L$ for almost  all $t\in[0,\alpha]$. So
$\zeta$ is a length minimizer parametrized by constant times arc-length.

We now use the fact that $\zeta(\alpha)\in U$, and let $\theta$ be 
a length minimizer 
joining $\zeta(\alpha)$ to $\hat x$. We parametrize $\theta$
so that its interval of definition is $[\alpha,1]$, and $\theta$ is
parametrized by constant times arc-length. Since each $\xi_j$ is a
minimizer, and $||\dot\xi_j(t)||_{_G}=L_j$ a.~e., we have 
$d_{{\cal M}}(\xi_j(\alpha),\hat x_j)=(1-\alpha)L_j$. So 
$d_{{\cal M}}(\zeta(\alpha),\hat x)=(1-\alpha)L$, and 
then $||\dot\theta(t)||_{_G}=L$ for a.~e. $t$. 

Now define $\xi_*:[0,1]\to M$ by letting $\xi_*(t)=\zeta(t)$ if
$t\in[0,\alpha]$ and $\xi_*(t)=\theta(t)$ if $t\in[\alpha,1]$. Then 
$\xi_*$ is an $E$-admissible $E$-arc such that $||\dot\xi_*(t)||_{_G}=L$
for a.~e. $t\in[0,1]$.  So $||\xi_*||_{_G}=L$. On the other hand, 
$$
d_{{\cal M}}(\xi_*(0),\xi_*(1))=d_{{\cal M}}(\bar x,\hat x)=
\lim_{j\to\infty}d_{{\cal M}}(\bar x_j,\hat x_j)=\lim_{j\to\infty}L_j=L\,.
$$
So $\xi_*$ is a length minimizer parametrized by constant timees arc-length. 
Therefore $\xi_*$ is a geodesic arc. 

Now write $\Xi_j=(\xi_j,\psi_j)$, and let $Z_j$ be the restriction
of $\Xi_j$ to the interval $[0,\alpha]$. Then the  $Z_j$ are in 
$C^0([0,\alpha],\pi_{_{T^*M}}^{-1}(B))\cap\widehat{GEO}({\cal M})$, 
and the projections $\zeta_j=\pi_{_{T^*M}}\compo Z_j$ are nonconstant 
length minimizers.  By Theorem \mainesti{}, all the $Z_j$ are contained in
a  fixed compact subset $J_B$ of $T^*M$. It then follows that we may
assume, after passing once more to a subsequence, that 
the limit $\bar p=\lim_{j\to\infty}\psi_j(0)$ exists.  

Let $\Xi\in\maxi{\widehat{GEO}({\cal M})}$ be such that $\Xi(0)=(\bar
x,\bar p)$. Then the domain of $\Xi$ is an interval $I$. We will show that
$[0,1]\subseteq I$.

Write $\Xi(t)=(\xi(t),\psi(t))$ for $t\in I$. 
Pick a $\beta$ such that $0<\beta\le\alpha$ and $[0,\beta]\subseteq I$. 
Then $\Xi_j(t)\to\Xi(t)$ for all $t\in[0,\beta]$, by the usual theorem 
on continuous dependence on the initial conditions of the solutions
of an ordinary differential equation. It follows that $\xi_j(t)\to\xi(t)$ 
for $t\in[0,\beta]$. Since $\xi_j(t)\to\xi_*(t)$ for $t\in[0,\alpha]$, we
conclude that $\xi$ and $\xi_*$ coincide on $[0,\beta]$. 

Since $\xi_*$ is a length minimizer parametrized by arc-length, 
it is the projection of an arc 
$\Xi_*\in C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$.
If $L\neq 0$, then Theorem \psiunique{} implies that 
$\Xi$ and $\Xi_*$ coincide on $[0,\beta]$, since $\xi$ and $\xi_*$ do. 
So $\Xi_*$ is a solution of the geodesic equation which is defined
on $[0,1]$ and satisfies $\Xi_*(0)=\Xi(0)$. Since $\Xi$ is maximally
defined, it follows that $[0,1]\subseteq I$. 
We now consider the remaining possibility, namely,  that $L=0$. In this
case $L_j\to 0$, and
$L_j^2=\sum_{i=1}^m\mylangle\psi_j(0),g_i(\xi_j(0))\myrangle^2$. It follows
that $\mylangle\bar p,g_i(\bar x)\myrangle=0$ for $i=1\ldots,m$. It is then 
easy to see that $(\bar x,\bar p)$ is an equilibrium point for the geodesic
equation, so $I=\rr$. 

We have therefore proved that the domain of $\Xi$ contains $[0,1]$. It then 
follows from the theorem 
on continuous dependence on the initial conditions of the solutions
of an ordinary differential equation that $\Xi_j(t)\to\Xi(t)$ uniformly
for $t\in[0,1]$. Therefore $(x_j,p_j)=\Xi_j(\tau_j)$ 
converges as $j\to\infty$ to the limit $\Xi(\tau)$. This completes 
the proof of (A'). \proofend

\vbox{

\rema
Theorem \bears{} is {\it not} valid in general ---even in the Riemannian
case--- if ($\#$) is replaced by the weaker requirement that 

{\sl

\itemitem{($\#_{w}$)} whenever $\bar x\in\bar K$ and $\hat x\in\hat K$, it
follows that there exists a length minimizer $\xi$ such that
$\partial\xi=(\bar x,\hat x)$.

}

\medbreak\noindent Indeed, we may take $M$ to be the two-dimensional unit
sphere  
from which one point $Q$ belonging to the Equator has been removed. 
Let $N$, $S$ be the North and South poles, respectively. Choose
$\bar K=\{S\}$, $\hat K=\{N\}$. Then ($\#_w$) holds, but ($\#$) does not. 
Clearly, the conclusion of Theorem \bears{} is not true. \proofend

}

\rema
In the Riemannian case, we can choose the set ${\cal K}$ of Theorem 
\bears{} to be the set of {\it all}
$\Xi\in C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$ 
whose projection $\pi_{_{T^*M}}\compo\Xi$ is 
in ${\cal EM}(\bar K,\hat K)$. {\it This does not work in the 
general subriemannian SBG case.} Indeed, if ${\cal M}$ is not Riemannian,
then we can pick a point $x\in M$ such that 
$\sigma_x$ is not surjective, and choose $\bar K=\hat K=\{x\}$. Then,
if $p\in E(x)^\perp$ is arbitrary, we can define 
$\Xi_{x,p}(t)=(x,p)$ for $t\in[0,1]$, and the arc $\Xi_{x,p}$ is in ${\cal
K}$.  Since $E(x)^\perp$ is not compact, we see that ${\cal K}$ is not compact
either. \proofend

\section{Subanalyticity of the distance function}

\noindent We are now ready to prove the result on subanalyticity of the
distance function $d_{{\cal M}}$.  Recall that, if $N_1,N_2$ are
real-analytic manifolds, and $\nu:N_1\to N_2$ is a mapping, then $\nu$ is
said to be {\it subanalytic} on a subset $S$ of $N_1$ if the set
$\{(x,\nu(x)):x\in S\}$ (i.~e. the graph of the restriction of $\nu$ to
$S$) is a subanalytic subset of $N_1\times N_2$.  We say that the
subriemannian manifold ${\cal M}=(M,F,\sigma,G)$ is {\it real-analytic} if
$M$ is a real-analytic manifold, $F$ is a real-analytic vector bundle over
$M$, and both the map $\sigma$ and the metric $G$ are real-analytic.

\xdef\subantheo{\the\sectno.\the\theono}
\theo
Let ${\cal M}=(M,F,\sigma,G)$ be a real-analytic strongly bracket-generating
subriemannian manifold.  Let $\bar K$, $\hat K$ be compact subanalytic
subsets of $M$ such that Condition ($\#$) of the statement of Theorem
\bears{} holds.  Then the distance function $d_{{\cal M}}$ is subanalytic
on $\bar K\times\hat K$.

\beginproof Using Theorem \bears{},   we let ${\cal K}$ be a compact subset
of $C^0([0,1],T^*M)\cap\widehat{GEO}({\cal M})$  such that
every $\xi\in{\cal EM}(\bar K,\hat K)$ is the
projection $\pi_{_{T^*M}}\compo\Xi$ of some $\Xi\in{\cal K}$.  We let
${\cal F}$ be the ``geodesic flow'' on $T^*M$, that is, the flow of the
vector field $\vec{H}_{{\cal M}}$. By definition, 

\medbreak\itemitem{(1)} ${\cal F}$ is a $T^*M$-valued partially defined map on
$T^*M\times\rr$;
\itemitem{(2)} a point $(x,p,t)\in{}T^*M\times\rr$ is in the domain ${\rm
Dom}({\cal F})$ of ${\cal F}$ iff the maximally defined solution
$\Xi_{x,p}$ of the initial value problem $\dot\Xi(s)=\vec{H}_{{\cal
M}}(\Xi(s))$, $\Xi(0)=(x,p)$, is defined for $s=t$;

\noindent and
\itemitem{(3)} ${\cal F}(x,p,t)=\Xi_{x,p}(t)$ if
$(x,p,t)\in{\rm{}Dom}({\cal{}F})$. 

\medbreak\noindent We then use ${\cal F}_1$ to denote the 
``exponential map,'' i.~e. the $T^*M$-valued partially defined map on
$T^*M$, whith domain 
${\rm{}Dom}({\cal{}F}_1)\,\,\equaldef\,\,\Big\{(x,p)\in{}T^*M:(x,p,1)
\in{\rm Dom}({\cal{}F})\Big\}$, defined by letting 
${\cal{}F}_1(x,p)\,\,\equaldef\,\,{\cal F}(x,p,1)$ for
$(x,p)\in{\rm Dom}({\cal F}_1)$.
Then ${\rm Dom}({\cal F}_1)$ is open in $T^*M$.   

Let $J=\{\Xi(0):\Xi\in{\cal K}\}$. Then $J$ is a compact subset of 
${\rm Dom}({\cal F}_1)$. Let $\tilde J$ be a compact subanalytic subset 
of ${\rm Dom}({\cal F}_1)$ such that $J\subseteq\tilde J$.  Let 
$A:{\rm Dom}({\cal F}_1)\to M\times M\times\rr$ be the map given by 
$A(x,p)=\Big(x\,,\,{\cal F}_1(x,p)\,,\,2H_{{\cal M}}(x,p)\Big)$. 
Then $A$ is a real-analytic map. Since $\tilde J$ is a compact subanalytic 
subset of ${\rm Dom}({\cal F}_1)$, the image $Q=A(\tilde J)$
is a compact subanalytic subset of $M\times M\times\rr$.  

Let 
$$
P=\Big\{(\bar x,\hat x,r)\!\in\! Q\cap(\bar K\!\times\!\hat K\!\times\!\rr)
:\,\sim(\exists s\!\in\!\rr)(0\!\le\!s\!<\!r\wedge (\bar x,\hat x,s)\!\in
\! Q\Big\}\,. 
$$
Then $P$ is a subanalytic subset of $M\times M\times\rr$.  ({\it Proof}:
\ This follows
from general properties of subanalytic sets, as explained, for example, 
in Sussmann [\sussdesin]: any formula of a first-order predicate calculus
constructed out of atomic formulae that define subanalytic sets, combined 
by means of the usual logical connectives  $\wedge$, $\vee$, $\sim$,
$\exists$ and $\forall$, automatically defines a subanalytic set, provided
only that all the quantifiers are proper. In our case, 
$$
(\bar x,\hat x,r)\!\in\!P\Longleftrightarrow\Biggl(
\bar x\!\in\!\bar K\wedge\hat x\!\in\!\hat K\wedge(\bar x,\hat x,r)
\!\in\!Q\wedge\sim(\exists s\!\in\!\rr)(0\!\le\! s\!<\!r\wedge
(\bar x,\hat x,s)\!\in\!Q) 
\Biggr)\,,
$$
and the existential quantifier is proper because of the bound 
$0\le s<r$.)

It is then easy to see that $P$ is precisely the graph 
of $d_{{\cal M}}\restr(\bar K \times\hat K)$.  So this graph 
is subanalytic, and our proof is complete. \proofend

Theorem \subantheo{} implies a number of subanalyticity results about the
distance function:

\xdef\issuban{\the\sectno.\the\theono}
\theo
Assume that ${\cal M}$ is SBG and real-analytic. Then 

{\sl 

\item{(1)} If $U$ is any open subset of $M\times M$ having the property that
for every pair $(\bar x,\hat x)\in U$ there exists a length minimizer 
$\xi$ such that $\partial\xi=(\bar x,\hat x)$, then $d_{{\cal M}}$ is
subanalytic on $U$ (that is, the graph of the restriction of $d_{{\cal M}}$
to $U$ is a subanalytic subset of $U\times\rr$);

\item{(2)} $d_{{\cal M}}$ is subanalytic on a neighborhood of the diagonal
of $M\times M$; 

\item{(3)} if $V$ is an open subset of $M$ which is relatively convex in
${\cal M}$ (cf. Definition \relcondef), then $d_{{\cal M}}$ is subanalytic 
on $V\times V$;

\item{(4)} if any two points of $M$ can be joined by a length minimizer,
then $d_{{\cal M}}$ is subanalytic on $M\times M$;

\item{(5)} if $M$ is complete,
then $d_{{\cal M}}$ is subanalytic on $M\times M$;

\item{(6)} if $\bar x\in M$ and $R>0$ are such that the closed ball
$B_R(\bar x)=\{x\in M:d_{{\cal M}}(x,\bar x)\}$ is compact, then 
the function $x\to d_{{\cal M}}(x,\bar x)$ is subanalytic 
on $B_R(\bar x)$. 

}

\beginproof Statement (1) follows from Theorem \subantheo{} because
subanalyticity is a local property: to prove that the graph $A=\{(\bar
x,\hat x,r)\in U\times\rr: r=d_{{\cal M}}(\bar x,\hat x)\}$ is subanalytic
in $U\times\rr$, it suffices to find, for each $(\bar x,\hat x,r)\in
U\times\rr$, an open neighborhood $W$ of $(\bar x,\hat x,r)$ in
$U\times\rr$ such that $A\cap W$ is subanalytic in $W$. If $r\neq d_{{\cal
M}}(\bar x,\hat x)$, then $A\cap W=\emptyset$ if $W$ is a sufficiently
small neighborhood of $(\bar x,\hat x,r)$, since $d_{{\cal M}}$ is
continuous.  If $r=d_{{\cal M}}(\bar x,\hat x)$, then we can pick compact
subanalytic subsets $\bar K$, $\hat K$, of $M$ such that $\bar K\times\hat
K\subseteq U$ and $\bar K\times\hat K$ is a neighborhood of $(\bar x,\hat
x)$. It is then clear that $\bar K$ has an open neighborhood $N$ such that
every pair $(x,x')\in N\times \hat K$ is the endpoint condition
$\partial\xi$ of some length minimizer $\xi$. So $\bar K$ and $\hat K$
satisfy the hypotheses of Theorem \subantheo, and then $A\cap(\bar
K\times\hat K\times\rr)$ is subanalytic in $M\times M\times\rr$. If $\bar
W$, $\hat W$ denote the interiors of $\bar K$, $\hat K$, and $W=\bar
W\times\hat W\times\rr$, then $W$ is a neighborhood of $(\bar x,\hat x,r)$
in $U\times\rr$, and $A\cap W$ is subanalytic in $W$. So a $W$
with the desired property exists, and then $A$ is subanalytic in
$U\times\rr$, completing the proof of (1).

Statement (2) is an immediate consequence of (1), because 
Proposition \eximinloc{} implies that there exists a neighborhood 
$U$ of the diagonal having the property of (1). 
Statement (3) also follows from (1), because if $V$ is relatively convex
then the set $U=V\times V$ satisfies the condition of (1). 
Statement (4) is a special case of (3). 
Statement (5) follows from (4) and the subriemannian Hopf-Rinow theorem
(Theorem \hopfrinow). 

Finally, to prove (6), we first observe that there exists $R'>R$ such that 
$B_{R'}(\bar x)$ is compact.  (Find finitely many compact balls 
$B_{r_\alpha}(x_\alpha)$, $\alpha\in A$, 
such that the balls $B_{{r_{\alpha}\over 2}}(x_{\alpha})$ 
cover $B_R(\bar x)$. Let 
$\rho=\min\Bigl({R\over 2}\,,\,\min\Big\{{r_{\alpha}\over 4}:\alpha\in
A\Big\}\Bigr)$, $R'=R+\rho$. 
If $x\in B_{R'}(\bar x)$, then there is an $E$-admissible $E$-curve
$\xi:[0,L]\to M$, parametrized by arc-length, such that $\xi(0)=\bar x$,
$\xi(L)=x$, and $||\xi||_G=L\le R'+\rho$. If $L\ge2\rho$, then
$x'=\xi(L-2\rho)\in B_R(\bar x)$, since $L-2\rho\le R$. So $x'\in
B_{{r_{\alpha}\over 2}}(x_{\alpha})$ for some $\alpha$, and then $x\in
B_{{r_{\alpha}}}(x_{\alpha})$, since $4\rho\le r_{\alpha}$. If $L< 2\rho$,
then $x\in B_R(\bar x)$. So $B_{R'}(\bar x)\subseteq \bigcup_{\alpha\in A}
B_{r_\alpha}(x_\alpha)$. Therefore 
$B_{R'}(\bar x)$ is compact.)

Now let $\beta={R'-R\over 2}$. If $d_{{\cal M}}(x,\bar x)<\beta$, 
then the closed ball $B_{R}(\bar x)$ is contained in 
$B_{R+\beta}(x)$, which is a closed subset of 
$B_{R'}(\bar x)$. So $B_{R+\beta}(x)$ is compact, and then Proposition 
\eximinloc{} tells us that every point of $B_{R+\beta}(x)$ can be joined to
$x$ by a minimizer.  Therefore 
every point of $B_{R}(\bar x)$ can be joined to
every point of $B_{\beta}(\bar x)$ by a minimizer. 
Therefore the hypothesis of Theorem \subantheo{}
is satisfied with $\bar K=\{\bar x\}$ and $\hat K=B_{R}(\bar x)$.
\proofend

\appen

\noindent{\bf Proof of Proposition \propinduces.}\ First, we show
that, if we define $\sigma_x(v)$ by $\sigma_x(v)=\sigma(V)(x)$, then
$\sigma_x(v)$ does not depend on the choice of $V$. If $V_1(x)=V_2(x)=v$,
then we can pick a local basis ${\bf f}=(f_1,\ldots,f_m)$ 
of sections of $F$ near $x$, and write
$W=V_1-V_2=\sum_{i=1}^mw^if_i$ on $U_{{\bf f}}$, where the $w^i$
belong to $C^\infty(U_{{\bf f}})$ and vanish at $x$. If $\rho\in
C^\infty(M)$ is such that $\rho(x)=1$ and ${\rm support}(\rho)$ is a
compact subset of $U_{{\bf f}}$, then
$$
\eqalign{
\sigma(W)(x)&=\rho(x)^2\sigma(W)(x)=\Bigl(\rho^2\sigma(W)\Bigr)(x)
=\sigma(\rho^2 W)(x)=\biggl(\sigma\Bigl(\sum_i(\rho w^i)(\rho
f_i)\Bigl)\biggl)(x)\cr
&=\biggl(\sum_i(\rho w^i)(\sigma(\rho f_i))\biggr)(x)
=\sum_i(\rho w^i)(x)\sigma(\rho f_i)(x)=0\,.\cr}
$$
So $\sigma(V_1)(x)=\sigma(V_2)(x)$, as desired.   
The last two statements are trivial.\proofend

\medbreak\noindent{\bf Proof of Proposition \propismeasur.} 
\ We let $n={\rm dim}(M)$, $m={\rm dim}(F)$, $I=I(\xi)$.  It clearly
suffices to work locally, i.e. to assume that $\xi(I)$ is entirely
contained in an open subset $U$ of $M$ which is the domain of a smooth
coordinate chart $\bx$ and of a local basis ${\bf
f}=(f_1,\ldots,f_m)$ of smooth sections of $F$.  Let $H\subseteq I$ be
such that $\dot\xi(t)$ exists and belongs to $E(\xi(t))$ for all
$t\in H$, and $I\backslash H$ has measure zero.  Let
$g_i(x)=\sigma_x(f_i(x))$.  For $t\in I$, define ${\cal G}(t)$ to be
the linear span of the vectors $g_1(\xi(t)),\ldots,g_m(\xi(t))$.
For $k\in\{1,\ldots,m\}$, and a subset $J$ of
$\{1,\ldots,m\}$ of cardinality $k$, let $I_{k,J}$ be the set of those
$t\in I$ such that ${\cal G}(t)$ has dimension $k$ and is spanned by
the vectors $g_j(\xi(t))$, $j\in J$.  Then $I_{k,J}$ is defined by a
finite collection of equalities and inequalities involving continuous
functions of $t$, namely, the $k\times k$ and $(k+1)\times(k+1)$
minors of the $m\times n$ matrix ${\bf g}(\xi(t))$ 
whose columns are the component
vectors of $g_1(\xi(t)),\ldots,g_m(\xi(t))$ relative to the chart
$\bx$.  So every $I_{k,J}$ 
is measurable. For $t\in I_{k,J}\cap H$, the vector $\dot\xi(t)$ can
be expressed uniquely as a linear combination
$\dot\xi(t)=\sum_{j\in J}\eta_{k,J}^j(t)g_j(\xi(t))$. 
It then follows easily that the functions $\eta_{k,J}^j$ are measurable,
since they can be obtained from the components of the vectors
$g_j(\xi(t))$ and $\dot\xi(t)$ by means of Cramer's rule.  So,
if we take $\eta_{k,J}(t)=\sum_{j\in J}\eta_{k,J}^j(t)f_j(\xi(t))$ for
$t\in I_{k,J}\cap H$, we see that $\eta_{k,J}$ is measurable on $I_{k,J}\cap
H$. Since $H=\cup_{k,J}(I_{k,J}\cap H)$, and $I\backslash H$ has measure
zero, the existence of a measurable $\zeta$
follows immediately. \proofend

\medbreak\noindent{\bf A.1. Example$\,$.}  
{\it (A smooth $E$-arc that is not $E$-admissible.)}   
\ Let $M=\rr^2$, and let $E$ be the singular distribution such that,
if $(x,y)\in M$, then  $E(x,y)$ is the linear span
of $X(x,y)$ and $Y(x,y)$, where $X=y{\partial\,\,\over\partial x}$
and $Y={\partial\,\,\over\partial y}$. (To realize $E$ as a singular
distribution, take $F$ to be the trivial bundle on $M$ with fiber $\rr^2$,
and let $\sigma_{(x,y)}(\alpha,\beta)=\alpha X(x,y)+\beta Y(x,y)$.) 
Let $\xi(t)=(t^2,t^2)$ for $t\in[-1,1]$. Then $\xi$ is a smooth $E$-arc,
and in fact $\xi$ is such that $\dot\xi(t)\in E(\xi(t))$ for {\it all}
(and not just {\it almost} all) $t\in[-1,1]$.
Let ${\bf f}=(f_1,f_2)$ be the canonical basis of sections of $F$.
Then $\sigma(f_1)=X$, $\sigma(f_2)=Y$, and 
the only expression of $\dot\xi(t)$ as a linear combination 
of $X(\xi(t))$ and $Y(\xi(t))$, for $t\neq 0$, is 
$$
\dot\xi(t)={2\over t}.X(\xi(t))+2t.Y(\xi(t))
\,\,\,\,{\rm for\,\,\,\,}t\in\rr\,.$$
This shows that $\xi$ is not $E$-admissible, since the function
$t\to{2\over t}$ is not integrable.  \proofend

\medbreak\noindent{\bf Proof of Proposition \propmeasmin{}}. \ We work
locally, as in the proof of Proposition 
\propismeasur. So we assume that $\xi$ is contained in the domain $U$ of a
chart $\bx$ and that a local orthonormal basis ${\bf
f}=(f_1,\ldots,f_m)$ of sections of $F$ exists on $U$. 
We let $Q$ be the set of those $t\in I(\xi)$ such that
$\dot\xi(t)$ exists, so $Q$ is measurable and $I\backslash Q$ is a
null set.   We define
${\cal G}(t)$, the $g_i$ and the sets $I_{k,J}$ exactly as in the
proof of Proposition \propismeasur, and we let $I_{k,J,0}$ be the set of those 
$t\in I_{k,J}$ such that $\dot\xi(t)$ exists and belongs to 
${\cal G}(t)$.  It is then clear that $I_0(\xi)$ is the union of the 
$I_{k,J,0}$ for all $k,J$.   

Fix $k$ and $J$, and choose a $t\in I_{k,J,0}$. Then we can find smooth vector
fields $Y_1,\ldots,Y_{n-k}$ on a neighborhood $V$ of $\xi(t)$ such
that $Y_1(x),\ldots,Y_{n-k}(x)$, together with the 
$g_j(x)$ for $j\in J$, form  a basis of $T_xM$ for all $x\in V$. 
We can then find $\eps>0$ such that
$\xi(\,]\,t-\eps,t+\eps\,[\,)\subseteq V$.  Then, if
$t-\eps<s<t+\eps$, we can write $\dot\xi(s)$ in a unique way 
as a linear combination 
$\dot\xi(s)=\sum_{i=1}^{n-k}\alpha^i(s)Y_i(\xi(s))+
\sum_{j\in J}\beta^j(s)g_j(\xi(s))$, 
and the coefficient functions $\alpha^i$, $\beta^j$ are measurable, 
because they are given, in terms of the components of $\dot\xi(s)$, 
the $Y_i(\xi(s))$, and the $g_j(\xi(s))$, by Cramer's rule. Clearly, 
$s\in I_{k,J,0}$ if and only if $s\in I_{k,J}$ and
$\alpha_1(s)=\ldots=\alpha_{n-k}(s)=0$. Since $I_{k,J}$ is measurable,
we conclude that $]t-\eps,t+\eps[\,\cap\,I_{k,J,0}$ is measurable.   

So we have shown  that for every $t\in I_{k,J,0}$ there is an
$\eps_t>0$ with the property that 
$\,]\,t-\eps_t,t+\eps_t\,[\,\cap\,I_{k,J,0}$ is
measurable.   Then the set 
$$
Z=\bigcup_{t\in I_{k,J,0}}\Bigl(\,]\,t-\eps_t,t+\eps_t\,[\,\cap\,
I_{k,J,0}\Bigr)
$$
is open in $\rr$, and $I_{k,J,0}$ is a locally measurable subset 
of $Z$. So $I_{k,J,0}$ is a measurable subset of $\rr$. 

It then follows that $I_0(\xi)$ is measurable. Next, we must show 
that $\omega_{\xi}$ is measurable on $I_0(\xi)$. It 
clearly suffices to fix $k$ and $J$, and show
that $\omega_{\xi}$ is measurable on $I_{k,J,0}$.  Since
$k$ and $J$ are now fixed, we may relabel the $f_i$'s so as to assume that 
$J=\{m-k+1,\ldots,m\}$.  Then for every $t\in I_{k,J,0}$ and
$i\in\{1,\ldots,m-k\}$ we can write
$g_i(\xi(t))=\sum_{j=m-k+1}^{m}v_{i}^{j}(t)g_j(\xi(t))$.  This expression
is unique (because the vectors $g_j(\xi(t))$, for $m-k+1\le j\le m$, are
linearly independent), and the functions $v_{i}^{j}$ are continuous
on $I_{k,J,0}$.  

If we define $\tilde
h_i(t)=f_i(\xi(t))-\sum_{j=m-k+1}^{m}v_{i}^{j}(t)f_j(\xi(t))$ for
$i=1,\ldots,m-k$, and $\tilde h_i(t)=f_i(\xi(t))$ for $i=m-k+1,\ldots,m$,
then the vectors $\tilde h_i(t)$ belong to $F(\xi(t))$ for every
$i\in\{1,\ldots,m\}$ and $t\in I_{k,J,0}$, and $(\tilde
h_1(t),\ldots,\tilde h_m(t))$ is a basis of $F(\xi(t))$ such that $(\tilde
h_1(t),\ldots,\tilde h_{m-k}(t))$ is a basis of the kernel $K(t)$ of
$\sigma_{\xi(t)}$.  Moreover, the $\tilde h_i$ depend continuously on $t$
for $t\in I_{k,J,0}$. If we apply the Gramm-Schmidt orthogonalization
procedure (with respect to the inner product $G(\xi(t))$ on $F(\xi(t))$) we
find an orthonormal basis $(h_1(t),\ldots,h_m(t))$ of $F(\xi(t))$ such that
the maps $t\to h_i(t)$ are continuous on $I_{k,J,0}$, and ${\rm
span}(h_1(t),\ldots,h_i(t))= {\rm span}(\tilde h_1(t),\ldots,\tilde
h_i(t))$ for every $t\in I_{k,J,0}$ and every $i\in\{1,\ldots,m\}$.  This
implies, in particular, that $(h_1(t),\ldots,h_{m-k}(t))$ is an orthonormal
basis of $K(t)$ and $(h_{m-k+1}(t),\ldots,h_{m}(t))$ is an orthonormal
basis of the orthogonal complement $K(t)^\perp$ of $K(t)$ in $F(\xi(t))$.

Let $h_i^{\#}(t)=\sigma_{\xi(t)}(h_i(t))$.  Then 
$h_1^{\#}(t)=\ldots=h_{m-k}^{\#}(t)=0$, and 
the vectors $h_{m-k+1}^{\#}(t),\ldots,h_{m}^{\#}(t)$ form a basis of
${\cal G}(t)$. 

For $t\in I_{k,J,0}\cap Q$, the vector $\dot\xi(t)$ belongs to ${\cal
G}(t)$, so it can be expressed in a unique way as a linear combination
$\dot\xi(t)=\sum_{j=m-k+1}^m\alpha^j(t)h_j^{\#}(t)$, and the
coefficient functions $\alpha^j$ are obviously measurable.  If we let
$\tilde\omega(t)=\sum_{j=m-k+1}^m\alpha^j(t)h_j(t)$, then $\tilde\omega$ is measurable
on $I_{k,J,0}\cap Q$ and $\sigma_{\xi(t)}(\tilde\omega(t))=\dot\xi(t)$.  If
$t\in I_{k,J,0}\cap Q$, and $w\in F(\xi(t))$ is such that
$\sigma_{\xi(t)}(w)=\dot\xi(t)$, then $w-\tilde\omega(t)\in K(t)^\perp$ and
$\tilde\omega(t)\in K(t)$, so $||w||_{_G}\ge||\tilde\omega(t)||_{_G}$. Therefore
$||\tilde\omega(t)||_{_G}=||\dot\xi(t)||_{_G}$. This shows that 
$\tilde\omega\equiv\omega_{\xi}$ on $I_{k,J,0}\cap Q$. So 
$\omega_{\xi}$ is measurable on $I_{k,J,0}\cap Q$, and our proof is
complete. \proofend

\medbreak\noindent{\bf Proof of Proposition \propisad{}.} 
\ The first assertion is trivial. To prove the second one,
we observe that, if $J\subseteq I(\xi)$ is a compact interval, then 
$\int_{J}||\omega_{\xi}(t)||_{_G}dt=\int_{J}||\dot\xi(t)||_{_G}dt$. If this
integral is finite for every 
choice of $J$, then $\eta_{{\bf f},\xi,J}$ is integrable whenever 
${\bf f}$ is an orthonormal local basis of sections of $F$ such 
that $\xi(J)\subseteq U_{{\bf f}}$. Therefore $\xi$ is
$E$-admissible. The converse is trivial, since the existence of some 
integrable control representation $\eta$ of $\xi$ on a compact subinterval
$J$ of $I(\xi)$ implies  that $||\omega_{\xi}||_{_G}$ is integrable on $J$,
because $\omega_{\xi}$ is measurable and 
$||\omega_{\xi}(t)||_{_G}\le||\eta(t)||$. \proofend

\xdef\propcompo{A.\the\theono}
\propa
Let $I_1$, $I_2$ be compact subintervals of $\rr$, and let 
$\theta:I_2\to I_1$ be absolutely
continuous, monotonically nondecreasing, and surjective. 
Let $h:I_1\to[0,\infty]$ be a Lebesgue-measurable nonnegative
extended-real-valued function. Then the function
$I_2\ni t\to h\bigl(\theta(t)\bigr).\dot\theta(t)\in[0,\infty]$
is Lebesgue measurable, and 
\xdef\equalh{A.\the\equano}
$$
\int_{I_1}h(s)\,ds=\int_{I_2}h\bigl(\theta(t)\bigr).\dot\theta(t)\,dt\,,\numeqa
$$
using, in the right-hand side, the convention  that $\infty.0=0$. 

\beginproof
We can assume without loss of generality that $h$ is bounded. Indeed, 
we can 
truncate $h$ by defining $h_N(t)=\min(h(t),N)$, so the $h_N$ are bounded.
Then $0\le h_N\uparrow h$ and $0\le (h_N\compo\theta).\dot\theta\uparrow
(h\compo\theta).\dot\theta$ pointwise as $N\uparrow\infty$, 
implying that $\int_{I_1}h_N\uparrow\int_{I_1}h$ and 
$\int_{I_2}(h_N\compo\theta).\dot\theta\uparrow
\int_{I_2}(h\compo\theta).\dot\theta$. (Notice that the statement 
that $(h_N\compo\theta).\dot\theta\uparrow(h\compo\theta).\dot\theta$
pointwise is valid {\it provided that we use the convention that
$\infty.0=0$}. ) 

Now, any bounded measurable function $h$ is a uniform limit of 
finite linear combinations of indicator functions of measurable sets, 
and the set of measurable functions for which (\equalh) holds is
obviously closed under uniform limits, since $\dot\theta$ is
integrable. So it suffices to prove that (\equalh) holds when 
$h$ is the indicator function $\chi_{_Q}$ of a Lebesgue-measurable subset
$Q$ of $I_1$. 

If $Q$ is a closed interval $[\alpha,\beta]$, then
(\equalh) holds. (Indeed, in this case $\int_{I_1}h=\beta-\alpha$,
and $\int_{I_2}(h\compo\theta).\dot\theta=
\int_{\theta^{-1}([\alpha,\beta])}\dot\theta(t)\,dt$.
Since $\theta$ is monotonically nondecreasing and surjective, 
the set $Z=\theta^{-1}([\alpha,\beta])$ is a closed interval
of the form $[\alpha',\beta']$, where 
$\alpha'$ and $\beta'$ are such that $\theta(\alpha')=\alpha$ and 
$\theta(\beta')=\beta$. But then
$$
\int_{\theta^{-1}([\alpha,\beta])}\dot\theta(t)\,dt=
\int_{\alpha'}^{\beta'}\dot\theta(t)\,dt=\theta(\beta')-\theta(\alpha')
=\beta-\alpha=\int_{I_1}h\,,$$
so (\equalh) holds, as stated.) 

It is clear that the set $S$ of all measurable $Q$ such that 
(\equalh) holds for $h=\chi_{_Q}$ is a monotone class, that is, a 
family of sets closed under increasing unions
and under decreasing unions. Since $S$ contains all compact
subintervals of $I_1$, it follows easily that $S$ contains 
all subintervals of $I_1$.  Moreover, $S$ is clearly closed under 
finite disjoint unions. So $S$ contains all finite unions of
intervals. It then follows from the Monotone Class Theorem (cf. [\halmos])
that $S$ contains all Borel measurable subsets of $I_1$. 
If $Q$ is an arbitrary Lebesgue measurable subset of $I_1$, then we
can write $Q=Q_1\cup Q_2$, with $Q_1$ Borel measurable, $Q_2$ a null
set, and $Q_1\cap Q_2=\emptyset$. Since $Q_1\in S$, it suffices to
prove that $Q_2\in S$.  We can find a Borel measurable subset 
$P$ of $I_2$ such that $Q_2\subseteq P$ and $P$ has measure zero. 
If $h$ is the indicator function of $P$, then we know 
that $\int_{I_1}h=\int_{I_2}(h\compo\theta).\dot\theta$. 
So $\int_{I_2}(h\compo\theta).\dot\theta=0$. Therefore 
$(h\compo\theta).\dot\theta$ vanishes almost everywhere. 
Let $h'$ be the indicator function of $Q_2$. Then $0\le h'\le h$
pointwise. So
$0\le(h'\compo\theta).\dot\theta\le(h\compo\theta).\dot\theta$ 
pointwise. Therefore 
$(h'\compo\theta).\dot\theta$ vanishes almost everywhere. So 
$\int_{I_2}(h'\compo\theta).\dot\theta=0=\int_{I_1}h'$. Therefore
$Q_2\in S$, and our proof is complete. \proofend


\xdef\propmapssets{A.\the\theono}
\propa
Let $I_1$, $I_2$ be compact subintervals of $\rr$, and let 
$\theta:I_2\to I_1$ be absolutely
continuous, monotonically nondecreasing, and surjective. 
Then (i)~if $S\subseteq I_2$ is Lebesgue measurable, it follows 
that $\theta(S)$ is Lebesgue measurable; (ii)~if $S$ has measure zero,
then $\theta(S)$ has measure zero; (iii)~if $B(\theta)$ is 
the set of all $t\in I_2$ such  that
$\dot\theta(t)$ exists and is $\neq 0$. Then 
$I_1\backslash\theta\Bigl(B(\theta)\Bigr)$ has measure zero. 

\beginproof
Suppose $S$ has measure zero. Let $\eps>0$.  Find $\delta>0$ such
that $\int_Q\dot\theta<\eps$ whenever $Q$ is measurable and such that 
${\rm meas}(Q)<\delta$. Then find a relatively open subset $U$ 
of $I_2$ such that ${\rm meas}(U)<\delta$ and $S\subseteq U$. Express
$U$ as the disjoint union of a sequence $\{I_j\}$ of relatively open
subintervals of $I_2$.  If $a_j,b_j$ are the left and right endpoints
of $I_j$, then
$\theta(S)\subseteq\theta(U)\subseteq\bigcup_j\,\,[\theta(a_j),\theta(b_j)]$.
So 
$$
{\rm meas}(\theta(S))\le{\rm meas}(\theta(U))
\le\sum_j(\theta(b_j)-\theta(a_j))=\int_U\dot\theta<\eps\,.
$$
Since $\eps$ was arbitrary, $\theta(S)$ has measure zero, and Part
(ii) is proved. 

Now, if $S\subseteq I_2$ is measurable, then for each positive 
integer $N$ we can find a compact subset $K_N$ of $S$ such that 
${\rm meas}(S\backslash K_N)<2^{-N}$.  If $K=\bigcup_NK_N$, then 
${\rm meas}(S\backslash K)=0$.  So 
${\rm meas}\Bigl(\theta(S\backslash K)\Bigr)=0$.  
Moreover, $\theta(S)=\theta(S\backslash K)\cup\theta(K) 
=\theta(S\backslash K)\bigcup\Bigl(\bigcup_N\theta(K_N) \Bigr)$.
Since the sets $\theta(K_N)$ are compact, we see that $\theta(S)$ is
the union of a set of measure zero and a countable family of compact
sets. So $S$ is measurable, and Part (i) is proved.  

To prove Part (iii), we let $Q=I_1\backslash\theta\Bigl(B(\theta)\Bigr)$,
and observe that $Q$ is measurable by Part (i). Let $h$ be 
the indicator function of $Q$. Then Proposition
\propcompo{} tells us that 
\xdef\vanishes{A.\the\equano}
$$
\int_{I_1}h(s)\,ds=\int_{I_2}h(\theta(t)).\dot\theta(t)\,dt\,.\numeqa
$$
Now, if $t\in I_1$ is such that $\dot\theta(t)$ exists and is $>0$,
then $t\in B(\theta)$, so $\theta(t)\in\theta\Bigl(B(\theta)\Bigr)$,
and  then  $\theta(t)\notin Q$, so $h(\theta(t))=0$, and 
$h(\theta(t))\dot\theta(t)=0$. Obviously, this last equality also holds 
if $\dot\theta(t)$ exists and is equal to $0$.  So 
$h(\theta(t))\dot\theta(t)=0$ for all $t$ for which 
$\dot\theta(t)=0$ exists, i.~e. for almost all $t\in I_2$. Therefore 
the right-hand side of (\vanishes) vanishes. So 
${\rm meas}(P)=\int_{I_1}h(s)\,ds=0$, 
and the proof of Part (iii) is complete. \proofend

\xdef\propcompomani{A.\the\theono}
\propa
Let $I_1$, $I_2$ be compact subintervals of $\rr$, and let 
$\theta:I_2\to I_1$ be absolutely
continuous, monotonically nondecreasing, and surjective. 
Let $M$ be a manifold, and let $\xi:I_1\to M$ be an absolutely
continuous arc. Let $\xi'=\xi\compo\theta$. Then $\xi'$ is absolutely
continuous  and
\xdef\manicomp{A.\the\equano}
$$
\dot\xi'(t)=\dot\xi(\theta(t)).\dot\theta(t)\numeqa
$$
for almost all $t\in I_2$, in the sense that, for almost every 
$t\in I_2$, $\dot\xi'(t)$ and $\dot\theta(t)$ exist, and 
either (a)~$\dot\theta(t)\neq 0$, $\dot\xi(\theta(t))$ exists, and 
(\manicomp) holds, or (b)~$\dot\theta(t)=0$ and
$\dot\xi'(t)=0$. \proofend

\beginproof
It is clear the result is local, so it suffices to assume 
that $M=\rr^n$. Then (\manicomp) is really a finite set of identities, 
one for each scalar component of the vector-valued functions 
$\xi$ and $\xi'$. So it suffices to assume that $n=1$. Let
$I_i=[a_i,b_i]$ for $i=1,2$. 

The function $\xi:I_1\to\rr$ is absolutely continuous, so we can write
$\xi=\xi_+-\xi_-$, where $\xi_+$ and $\xi_-$ are absolutely continuous
and monotonically nondecreasing.  (For example, we may take 
$\xi_+(s)=\int_{a_1}^s|\dot\xi(r)|\,dr$.)  Let
$\xi_-'=\xi_-\compo\theta$, $\xi_+'=\xi_+\compo\theta$. Then 
$\xi'=\xi_+'-\xi_-'$. 

For $s\in I_1$, let 
$$
\eqalign{
h_+(s)&=\dot\xi_+(s)
\,\,\,{\rm if}\,\,\,\dot\xi_+(s)\,\,\,{\rm  exists}\,,\qquad
h_+(s)=+\infty\,\,\,{\rm if}\,\,\,\dot\xi_+(s)\,\,\,{\rm
  does\,\,\,not\,\,\, 
  exist}\,,\cr
h_-(s)&=\dot\xi_-(s)
\,\,\,{\rm if}\,\,\,\dot\xi_-(s)\,\,\,{\rm  exists}\,,\qquad
h_-(s)=+\infty\,\,\,{\rm if}\,\,\,\dot\xi_-(s)\,\,\,{\rm
  does\,\,\,not\,\,\, 
  exist}
\,.\cr}
$$
Then $h_+$ and $h_-$ are Lebesgue measurable $[0,\infty]$-valued 
functions defined everywhere on $I_1$, so we can conclude from
Proposition \propcompo{}  that the nonnegative functions 
$t\to h_+(\theta(t)).\dot\theta(t)$ and $t\to
h_-(\theta(t)).\dot\theta(t)$ are Lebesgue-measurable on $I_2$. 
Moreover, Proposition \propcompo{} also yields, 
when applied to the interval $[a_2,t]$ for any $t\in[a_2,b_2]$, the
identities
\xdef\identia{A.\the\equano}
$$
\int_{a_1}^{\theta(t)}h_+(s)\,ds=\int_{a_2}^th_+(\theta(\tau)).\dot\theta(\tau)
\,d\tau\,,\numeqa
$$
\xdef\identib{A.\the\equano}
$$
\int_{a_1}^{\theta(t)}h_-(s)\,ds=\int_{a_2}^th_-(\theta(\tau)).\dot\theta(\tau)
\,d\tau\,.\numeqa
$$
In particular, (\identia) and (\identib) imply that
$\int_{a_2}^{b_2}h_+(\theta(\tau)).\dot\theta(\tau)\,d\tau=
\xi_+(b_1)-\xi_+(a_1)<\infty$
and 
$\int_{a_2}^{b_2}h_-(\theta(\tau)).\dot\theta(\tau)\,d\tau=\xi_-(b_1)-\xi_-(a_1)<\infty$.
Therefore the nonnegative measurable functions  
$t\to h_+(\theta(t)).\dot\theta(t)$ and $t\to
h_-(\theta(t)).\dot\theta(t)$ are integrable.
Then (\identia) implies that
$$
\eqalign{
\xi_+'(t)-\xi_+'(a_2)&=\xi_+(\theta(t))-\xi_+(\theta(a_2))
=\xi_+(\theta(t))-\xi_+(a_1)
=\int_{a_1}^{\theta(t)}\dot\xi_+(s)\,ds\cr
&=\int_{a_1}^{\theta(t)}h_+(s)\,ds
=\int_{a_2}^th_+(\theta(\tau)).\dot\theta(\tau)\,d\tau\,.\cr}
$$
So $\xi_+'$ is the indefinite integral of an integrable function, and
therefore $\xi_+'$ is absolutely continuous.   A similar argument,
using (\identib), shows that 
$$
\xi_-'(t)-\xi_-'(a_2)=\int_{a_2}^th_-(\theta(\tau)).\dot\theta(\tau)\,d\tau
$$
for all $t\in I_2$, so $\xi_-'$ is absolutely continuous as well. 
It then follows that  $\xi'$ is absolutely continuous.

The Lebesgue differentiation theorem then implies that there is a
subset $Z$ of $I_2$ such that $I_2\backslash Z$ has zero
Lebesgue measure, having the properties that, for all $t\in Z$: 

\medbreak
\itemitem{(i)} {\it $\dot\xi_+'(t)$ and $\dot\xi_-'(t)$ exist 
and  are equal to $h_+(\theta(t)).\dot\theta(t)$ and   
$h_-(\theta(t)).\dot\theta(t)$, respectively,}

\medbreak
\itemitem{(ii)} {\it $\dot\theta(t)$ exists}.

\medbreak\noindent
Now let $t\in Z$. Then the products $h_+(\theta(t)).\dot\theta(t)$
and $h_-(\theta(t)).\dot\theta(t)$ are finite. Suppose first that
$\dot\theta(t)\neq 0$. Then $h_+(\theta(t))$ and $h_-(\theta(t))$ must
themselves be finite. It then
follows from our definition of $h_+$ and $h_-$ that 
$\dot\xi(t)$ exists and is equal to 
$h_+(\theta(t))-h_-(\theta(t))$. Moreover, (i) implies that 
$\dot\xi'(t)$ exists and is equal to 
$\Big(h_+(\theta(t))-h_-(\theta(t))\Big).\dot\theta(t)$, i.~e. to
$\dot\xi(t).\dot\theta(t)$. So (\manicomp) holds. 

Next consider the case when $\dot\theta(t)=0$. In this case, we are
not guaranteed that $h_+(\theta(t))$ and   
$h_-(\theta(t))$ are finite, but it is still true 
that $h_+(\theta(t)).\dot\theta(t)=h_-(\theta(t)).\dot\theta(t)=0$. 
Then (i) implies that $\dot\xi'(t)=0$ as well. \proofend

\remaa
{\it It is not possible to strengthen the conclusion of Proposition 
\propcompomani{} to obtain a subset $Z$ of $I_2$ of full measure, such 
that $\dot\xi'(t)$, $\dot\xi(\theta(t))$ and $\dot\theta(t)$ exist
for all $t\in Z$.} For example, $\theta$ could be constant on a nontrivial
subinterval $J$ of $I_2$, and its value on $J$ could be a point of 
$I_1$ where $\xi$ is not differentiable. Then the set of points $t$
such that $\dot\xi(\theta(t))$ does not exist will have positive
measure. \proofend

\medbreak\noindent{\bf Proof of Proposition \propinvarep{}.}
\ Let $I_i=I(\xi_i)$. Let $\theta:I_2\to I_1$ be absolutely
continuous, monotonically nondecreasing, surjective, and such that
$\xi_1\compo\theta=\xi_2$.     Then Proposition
\propcompomani{} tells us that there is a subset $Z$ of $I_2$ of full
measure such that, if $t\in Z$, then 
(a)~$\dot\xi_2(t)$ and $\dot\theta(t)$ exist, 
(b)~$\dot\xi_2(t)$ vanishes if $\dot\theta(t)$ does, and
(c)~if $\dot\theta(t)\neq 0$ then 
$\dot\xi_1(\theta(t))$ exists and satisfies 
$\dot\xi_2(t)=\dot\xi_1(\theta(t)).\dot\theta(t)$.

If we let $h(s)=||\dot\xi_1(s)||_{_G}$ when $\dot\xi_1(s)$ exists, and 
$h(s)=+\infty$ otherwise, then 
\xdef\xionetwo{\equanosectno}
$$
||\dot\xi_2(t)||_{_G}=h(\theta(t)).\dot\theta(t) \numeq
$$
for all $t\in Z$, if we make the convention that $\infty.0=0$, as in
the statement of Proposition \propcompo. 
So $||\xi_1||_{_G}=\int_{I_1}h=\int_{I_2}(h\compo\theta).\dot\theta
=\int_{I_2}||\dot\xi_2(t)||_{_G}\,dt=||\xi_2||_{_G}$,
using Proposition \propcompo{} and Formula (\xionetwo). \proofend

\medbreak\noindent{\bf Proof of Proposition \propunique{}.}
\ We let $I(\xi)=[a,b]$, and define $L=||\xi||_{_G}$. 

We first prove uniqueness. Let $\xi_1$, $\xi_1'$ be two members of 
$TR(\xi)\cap{\cal{A}}^{PAL,0}({\cal M})$. Then 
$||\xi_1||_{_G}=||\xi_1'||_{_G}=L$ by Proposition \propinvarep, 
so the intervals $I(\xi_1)$ and $I(\xi_1')$ are both equal to $[0,L]$
by (\itiscle). 

Let $\theta:I(\xi)\to[0,L]$ and $\theta':I(\xi)\to[0,L]$ be absolutely
continuous, nondecreasing, surjective, and such that 
$\xi_1\compo\theta=\xi_1'\compo\theta'=\xi$. 
If $t\in I(\xi)$ is arbitrary, use 
$\xi_t$, $\xi_{1,t}$, $\xi_{1,t}'$ to denote, respectively,
the restrictions of $\xi$, $\xi_1$ and $\xi_1'$ to the intervals
$[a,t]$, $[0,\theta(t)]$ and $[0,\theta'(t)]$. Also, let 
$\theta_t$ and $\theta_t'$ be the restrictions of $\theta$ and
$\theta'$ to $[a,t]$. Then $\xi_{1,t}$ and $\xi_{1,t}'$ are both 
members of $TR(\xi_t)\cap{\cal{A}}^{PAL,0}({\cal M})$. So we already 
know that their domains $[0,\theta(t)]$ and $[0,\theta'(t)]$
coincide. Therefore $\theta(t)=\theta'(t)$.  

We have thus shown that $\theta(t)=\theta'(t)$ for all $t\in[a,b]$. 
If $s\in[0,L]$, then we can find $t\in[a,b]$ such that
$\theta(t)=s$. Then $\theta'(t)=s$ as well. So
$\xi_1(s)=\xi_1(\theta(t))=\xi(t)$, and 
$\xi_1'(s)=\xi_1'(\theta'(t))=\xi(t)$. Therefore $\xi_1\equiv\xi_1'$,
proving the uniqueness of $\xi_1$.  
The uniqueness of $\xi_2$ then follows trivially.
  
The existence of $\xi_2$ follows easily from 
the corresponding assertion for $\xi_1$ if $||\xi||_{_G}>0$, and is
proved directly if $||\xi||_{_G}=0$ but $\xi$ is nontrivial. 

To prove the existence of $\xi_1$, we first observe that 
the conclusion is valid for a finite concatenation of curves if it is
valid for each of them. Therefore we can work locally, that is, we can
assume that $M=\rr^n$ and there exists a global orthonormal basis 
${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$. We let
$g_i=\sigma(f_i)$. 

We then  define a map $\theta:[a,b]\to[0,L]$ by letting
$\theta(t)=\int_{a}^t||\dot\xi(\tau)||_{_G}\,d\tau$.
Then $\theta$ is absolutely continuous, because the function
$t\to||\dot\xi(t)||_{_G}$ is integrable. Moreover, $\theta$ is obviously
monotonically nondecreasing and surjective.  

If  $a\le t_1< t_2\le b$ and $\theta(t_2)=\theta(t_1)$,
then $\int_{t_1}^{t_2}||\dot\xi(s)||_{_G}ds=0$, so 
$\|\dot\xi(t)\|_{_G}=0$ for almost all $t\in[t_1,t_2]$, and then
$\dot\xi(t)=0$ for almost all $t\in[t_1,t_2]$ by Remark \remisnul.
Therefore $\xi(t_2)=\xi(t_1)$. This enables us to define $\xi_1:[0,L]\to M$
by letting $\xi_1(s)=\xi(t)$ if $\theta(t)=s$. It is then clear that
$\xi_1\compo\theta=\xi$.  

Let 
$C=\sup\,\Bigl\{\Big(\sum_{i=1}^m||g_i(\xi(t))||^2\Big)^{{1\over
    2}}:t\in[a,b]\Bigr\}$.
Let $\eta=\eta_{{\bf f},\xi,[a,b]}$. 
If  $0\le s_1< s_2\le L$, and $s_i=\theta(t_i)$, then
$$
\eqalign{
||\,\xi_1(s_2)-\xi_1(s_1)\,||&=||\,\xi(t_2)-\xi(t_1)\,||
=||\int_{t_1}^{t_2}\dot\xi(t)\,dt||
=||\int_{t_1}^{t_2}\Bigl(\sum_{i=1}^m\eta^i(t)g_i(\xi(t))\Bigr)\,dt\,||\cr
&\le C\int_{t_1}^{t_2}||\eta(t)||\,dt
=C\Big(\theta(t_2)-\theta(t_1)\Big)
=C(s_2-s_1)\,,\cr}
$$
so $\xi_1$ is a Lipschitz arc. 

To conclude the proof, we have to show that $\xi_1\in{\cal A}_E(M)$,
i.e. that $\dot\xi_1(s)\in E(\xi_1(s))$ for almost every $s\in I_1$.
Let $B_1$ denote the set of all points $t\in [a,b]$ such that $\xi$
and $\theta$ are differentiable at $t$ and $\dot\xi(t)\in E(\xi(t))$.
Then $[a,b]\backslash B_1$ has measure zero. Let $B$ be the set of
those $t\in[a,b]$ such that $\dot\theta(t)$ exists and is not equal to
zero. Then Proposition \propmapssets{} tells us that
$[0,L]\backslash\theta(B)$ has measure zero. In addition,
$B\backslash B_1$ has measure zero, so Proposition \propmapssets{}
also implies that $\theta(B\backslash B_1)$ has measure zero. 
Let $H_1$  be the set of those $s\in[0,L]$ such that the derivative
$\dot\xi_1(s)$ does not exist. Then ${\rm meas}(H_1)=0$, because $\xi_1$ is
Lipschitz.  Define 
$H=H_1\cup\Big([0,L]\backslash\theta(B)\Big)\cup\theta(B\backslash B_1)$.
Then  ${\rm meas}(H)=0$. Suppose $s\in[0,L]$ and $s\notin H$.  
Then $\dot\xi_1(s)$ exists, because $s\notin H_1$. Also, 
$s$ belongs to $\theta(B)$, so $s=\theta(t)$ for some $t\in B$. Then
$t$ must belong to $B_1$, for otherwise $s$ would be in 
$\theta(B\backslash B_1)$.   Therefore $\xi$ and $\theta$ are
differentiable at $t$, and $\dot\xi(t)\in E(\xi(t))$. Since
$\xi=\xi_1\compo\theta$, and $\xi_1$ is differentiable at 
$s=\theta(t)$, the chain rule implies that
$\dot\xi(t)=\dot\xi_1(s).\dot\theta(t)$. Since $\dot\xi(t)\in
E(\xi(t))$,  and $\dot\theta(t)\neq 0$ (because $t\in B$), it follows
that $\dot\xi_1(s)\in E(\xi(t))=E(\xi_1(s))$.
Since this is true for all $s\in[0,L]\backslash H$, and $H$ has 
measure zero, our proof is complete. \proofend

\medbreak\noindent{\bf Proof of Proposition \proplipimplies{}.}
\ It is clear
that $(b)\Rightarrow(a)$. To prove the reverse implication, it
suffices to work locally. So we assume that $M=\rr^n$ and
there exists a global orthonormal basis ${\bf f}=(f_1,\ldots,f_m)$ of
sections of $F$. We let $I(\xi)=[a,b]$, and write $g_i=\sigma(f_i)$.  

For each positive integer $N$, let $t_{k,N}=a+k2^{-N}(b-a)$, for 
$k=0,1,\ldots,2^N$.   Let $x_{k,N}=\xi(t_{k,N})$. Then for each 
$k\in\{1,\ldots,2^N\}$ 
the distance $d_{{\cal M}}(x_{k-1,N},x_{k,N})$ is less than 
$c(b-a)2^{-N}$, so we may find an $E$-admissible $E$-arc 
$\xi_{k,N}:[t_{k-1,N},t_{k,N}]\to\rr^n$ that satisfies
$$
\xi_{k-1,N}(t_{k-1,N})=x_{k-1,N}\,,\quad
\xi_{k,N}(t_{k,N})=x_{k,N}\,,\quad
||\xi_{k,N}||_{_G}\le c(b-a)2^{-N}(1+2^{-N})\,,
$$
and is parametrized by constant times arc-length.   

We then find a control representation
$\eta_{k,N}:[t_{k-1,N},t_{k,N}]\to\rr^m$ of $\xi_{k,N}$, so 
$$
\dot\xi_{k,N}(t)=\sum_{i=1}^m\eta_{k,N}^i(t)g_i(\xi_{k,N}(t))\quad{\rm
  for\,\,\,a.~e.\quad}t\in[t_{k-1,N},t_{k,N}]\,,
$$
and $||\eta_{k,N}(t)||=c_{k,N}$ for a.~e. $t\in[t_{k-1,N},t_{k,N}]$, where
the constant $c_{k,N}$ is such that
$2^{-N}(b-a)c_{k,N}=||\xi_{k,N}||_{_G}$. Since 
$||\xi_{k,N}||_{_G}\le c(b-a)2^{-N}(1+2^{-N})$,  it follows that 
$c_{k,N}$ satisfies $c_{k,N}\le c(1+2^{-N})$. 

Next, define $\xi_N:[a,b]\to \rr^n$, $\eta_N:[a,b]\to \rr^m$, 
by letting 
$$
\xi_N(t)=\xi_{k,N}(t)\quad{\rm and}\quad 
\eta_N(t)=\eta_{k,N}(t)\quad{\rm whenever}\quad
t\in[t_{k-1,N},t_{k,N}]\,.
$$
Then each $\xi_N$ is a trajectory for the control $\eta_N$, and
$\eta_N$ satisfies the pointwise bound $||\eta_N(t)||\le 
c(1+2^{-N})$ for a.~e. $t$. It follows that the set
$\{\eta_N:N\in\nn\}$ is weak$^*$-precompact in
$L^{\infty}([a,b],\rr^m)$. So there exists an infinite subset 
$J$ of $\nn$, such that the limit
$\eta_J={\rm weak}^*$-$\lim_{N\to_J\infty}\eta_N$
exists. Then the curves $\xi_N$, $N\in J$, converge
uniformly to a limit $\xi_J$, which satisfies 
$\xi_J(t)=\xi_J(a)+\int_a^t\Big(\sum_{i=1}^m\eta_J^i(\tau)g_i
(\xi_J(\tau))\Big)\,d\tau$ for $t\in[a,b]$. 

On the other hand, $\xi_{N}(t_{k,N})=x_{k,N}=\xi(t_{k,N})$ by
construction. Given any $t\in[a,b]$ which is of the form $t_{k,N}$ for
some $N,k$, we also have $t=t_{k2^{\nu-N},\nu}$ for all $\nu$
such that $\nu\ge N$.  This happens, in particular, for infinitely
many values of $\nu$ belonging to $J$. So $\xi_\nu(t)=\xi(t)$ for
infinitely many $\nu$'s in $J$, and then $\xi_J(t)=\xi(t)$.  Since
this happens for all $t$'s in a dense subset of $[a,b]$, we can conclude
that $\xi_J\equiv \xi$. So $\xi$ is in  fact an $E$-admissible $E$-arc,
having a control representation with control $\eta_J$. It is clear
that $||\eta_J(t)||\le c$ a.~e., so $||\dot\xi(t)||_{_G}\le c$ a.~e., and
our conclusion is proved. \proofend

\medbreak\noindent{\bf Proof of Proposition \equivlengen{}.} \ Let 
$\xi\in{\cal A}_{[a,b}({\cal{}M})$, and write 
$(\bar x,\hat x)=\partial\xi$.   If $\xi$ is not an
$E$-curve, then $||\dot\xi(t)||_{_G}=\infty$ for $t$ in a set of positive
measure, so both $||\xi||_{_G}$ and ${\cal E}_{{\cal M}}(\xi)$ are
infinite.  If $\xi$ is an $E$-curve but isn't admissible, then 
$\int_a^b||\dot\xi(t)||_{_G}dt=+\infty$.  Since 
$||\dot\xi(t)||_{_G}^2\ge||\dot\xi(t)||_{_G}-1$ for all $t$, we have 
$\int_a^b||\dot\xi(t)||_{_G}^2dt=+\infty$ as well. 

Now suppose $\xi$ is $E$-admissible. Let $\xi_1$ be its canonical
reparametrization, and define 
$\xi_2(t)=\xi_1\Bigl({t-a\over b-a}\Bigr)$ for $a\le t\le b$.
Then the function $t\to||\dot\xi_2(t)||_{_G}$ is equal a.~e.{} to a constant
$c$, and $||\xi||_{_G}=||\xi_2||_{_G}=(b-a).c$.  Therefore
\xdef\stringa{A.\the\equano}
$$
\eqalign{
d_{{\cal M}}(\bar x,\hat x)\le||\xi||_{_G}
&=\int_a^b||\dot\xi(t)||_{_G}dt
\le(b-a)^{1/2}.\Big(\int_a^b||\dot\xi(t)||_{_G}^2dt\Big)^{1/2}\cr
&=(b-a)^{1/2}{\cal E}_{{\cal M},a,b}(\xi)^{1/2}\,,\cr}\numeqa
$$
and
\xdef\stringb{A.\the\equano}
$$
\eqalign{
(b-a).d^{\,{\cal E},a,b}(\bar x,\hat x)
&\le(b-a).{\cal E}_{{\cal M},a,b}(\xi_2)
=(b-a).\int_a^b||\dot\xi_2(t)||_{_G}^2dt\cr
&=(b-a)^2c^2=||\xi_2||_{_G}^2=||\xi||_{_G}^2\,.\cr}\numeqa
$$
In (\stringa), we can  take the infimum with respect to 
all $\xi$ such that $\partial\xi=(\bar x,\hat x)$, and conclude that 
$d_{{\cal M}}(\bar x,\hat x)\le (b-a)^{1/2}d^{\,{\cal E},a,b}(\bar x,\hat
x)^{1/2}$.  
If, on the other hand, we take the infimum in (\stringb),
we get 
$d_{{\cal M}}(\bar x,\hat x)^2\ge (b-a)^{1/2}d^{\,{\cal E},a,b}(\bar x,\hat
x)$.  This proves Part (A).

To prove (B), we observe that, if $\xi$ is an energy minimizer, then   
$$
(b-a)^{1/2}.{\cal E}_{{\cal M},a,b}(\xi)^{1/2}=
(b-a)^{1/2}.d^{\,{\cal E},a,b}(\bar x,\hat x)^{1/2}=
d_{{\cal M}}(\bar x,\hat x)\,,
$$
so the inequalities of (\stringa) are in fact equalities. Therefore
$d_{{\cal M}}(\bar x,\hat x)=||\xi||_{_G}$ ---showing that $\xi$ is a
length minimizer--- and 
$\int_a^b||\dot\xi(t)||_{_G}dt=(b-a)^{1/2}\Big(\int_a^b||
\dot\xi(t)||_{_G}^2\,dt\Big)^{1/2}$, 
which, as is well known, implies that the function 
$t\to||\dot\xi(t)||_{_G}$ is a.~e. constant. So
$\xi$ is parametrized by constant times arc-length, and the implication
(B.1)$\Rightarrow$(B.2) has been proved. 

Conversely, suppose $\xi$ is a length minimizer parametrized by constant
times arc-length.  Then $\xi=\xi_2$, and 
$||\xi||_{_G}^2=d_{{\cal{}M}}(\bar x,\hat x)^2=
(b-a).d^{\,{\cal E},a,b}(\bar x,\hat x)$.
So the inequality of (\stringb) is in fact an equality. Therefore
$d^{\,{\cal E},a,b}(\bar x,\hat x)={\cal E}_{{\cal M},a,b}(\xi_2)=
{\cal E}_{{\cal M},a,b}(\xi)$, 
showing that $\xi$ is an energy minimizer. \proofend

\medbreak\noindent{\bf Proof of the assertions of Example \extexamp{}.}
\ Suppose $\xi\in\maxi{{\cal{}C}^{PAL}({\cal M})}$ and 
$\tau$ is a limit point of $I(\xi)$ such that $\bar
x=\lim_{t\to_{_{I(\xi)}}\tau}\xi(t)$ exists. Then $\tau\in I(\xi)$, for
otherwise we would be able to extend $\xi$ to $I(\xi)\cup\{\tau\}$ and
obtain a new curve in ${\cal{}C}^{PAL}({\cal M})$ strictly extending $\xi$,
thereby contradicting the maximality of $\xi$.

Find an orthonormal basis ${\bf f}=(f_1,\ldots,f_m)$ of
sections of $F$ in a neighborhood of $\bar x$, and write
$g_i=\sigma(f_i)$.   Then the maximality of 
$\xi$ implies that $I(\xi)$ is nontrivial. Since
$\tau\in I(\xi)$, there is an $\eps>0$ such that 
at least one of the intervals
$A_-=\,]\,\tau-\eps,\tau\,]$, $A_+=\,[\,\tau,\tau+\eps\,[$, is 
contained in $I(\xi)$.
By making $\eps$ smaller, if necessary, we may assume that 
$\xi\Big((A_-\cup A_+)\cap I(\xi)\Big)\subseteq U_{{\bf f}}$.

Suppose, for example, that $A_-\subseteq I(\xi)$, Choose an integrable
control $\eta:A_-\to\rr^m$ such that
$\dot\xi(t)=\sum_i\eta^i(t)g_i(\xi(t))$ and
$||\dot\xi(t)||_{_G}=||\eta(t)||=1$ for a.~e. $t\in A_-$.  Extend $\eta$
to an integrable function $\tilde\eta: A_-\cup A_+\to\rr^m$ such that
$||\tilde\eta(t)||=1$ for a.~e. $t\in A_-\cup A_+$.  
By the existence theorem for solutions of ordinary
differential equations, there exists ---after making $\eps$ smaller,
if necessary--- a solution $\xi_1:A_-\cup A_+\to
M$ of the initial value problem $\dot
x(t)=\sum_i\tilde\eta^i(t)g_i(x(t))$, $x(\tau)=\xi(b)$.   
Then $\xi_1=\xi$ on $A_-$. If we reparametrize $\xi_1\restr A_+$ by  
arc-length,  and then concatenate the result with $\xi$, we obtain 
a  curve $\xi'\in{\cal C}^{PAL}({\cal M})$ that extends $\xi$ to a
strictly larger interval, contradicting the maximality of $\xi$. 

The proof that ${\cal C}^{a,\le c}({\cal M})$ has the extension property is
identical to the above argument, except that now $\eta$ and $\tilde\eta$
are chosen so that $||\eta(t)||\le c$ and $||\tilde\eta(t)||\le c$ almost
everywhere, and there is no need to reparametrize $\xi_1\restr A_+$. \proofend

\medbreak\noindent{\bf Proof of Proposition \compaprop.}\ The 
set ${\cal A}^{a,\le  c}({\cal M})$  is an {\it arc
  system} on the metric space $(M,d_{{\cal M}})$, with time domain
  $\rr$, in the sense of [\liuss]. Then
${\cal C}^{a,\le  c}({\cal M})$  is exactly the set of all curves 
$\xi$ in $M$ such that the restriction $\xi\restr J$ of $\xi$ to every
  compact subinterval of $I(\xi)$ is in ${\cal A}^{a,\le  c}({\cal M})$.
That is,  ${\cal C}^{a,\le  c}({\cal M})=\Big({\cal A}^{a,\le  c}({\cal
  M})\Big)^{\#}$, using the  notations of [\liuss]. 

Moreover, ${\cal A}^{a,\le  c}({\cal M})$  has the
``compactness property'' of [\liuss]. Indeed, if
$\{\xi_j\}_{j=1}^\infty$ is a sequence of arcs in 
${\cal A}^{a,\le  c}({\cal M})$, whose graphs
$G(\xi_j)\,\,\equaldef\,\,\Big\{\,(\xi_j(t),t):t\in I(\xi_j)\,\Big\}$
are all contained in
some fixed compact subset $K$ of $M\times\rr$, then there is an
infinite subset $S$ of $\nn$ such that, if $I(\xi_j)=[a_j,b_j]$, then
the limits $a=\lim_{j\to_S\infty}a_j$ and 
$b=\lim_{j\to_S\infty}b_j$ exist.  Since the $\xi_j$ belong to 
${\cal A}^{a,\le  c}({\cal M})$,  the inequality 
$d_{{\cal M}}(\xi_j(t),\xi_j(t'))\le c|t-t'|$ holds whenever 
$t,t'\in[a_j,b_j]$, so the sequence 
$\{\xi_j\}_{j=1}^\infty$ is equicontinuous. Since all the $\xi_j$ are
contained in a fixed compact subset of $M$, the Ascoli-Arzel\`a
Theorem implies that there is an infinite subset $S'$ of $S$ such that
the arcs $\xi_j$, $j\in S'$, converge uniformly (in the sense that,
if $a_j\le t_j\le b_j$ and $t_j\to t$ then $\xi_j(t_j)\to\xi(t)$) 
to an arc $\xi:[a,b]\to M$. Then the inequality
$d_{{\cal M}}(\xi(t),\xi(t'))\le c|t-t'|$ holds whenever 
$t,t'\in[a,b]$. Therefore Proposition \proplipimplies{}
implies that $\xi\in{\cal A}^{a,\le  c}({\cal M})$.   

Moreover, is t is clear that ${\cal A}^{a,\le  c}({\cal M})$  also has
the ``local existence property'' of [\liuss]. So 
${\cal A}^{a,\le  c}({\cal M})$ satisfies the hypotheses of 
Proposition 2 of [\liuss].  The conclusion then follows from that
proposition. \proofend 

\medbreak\noindent{\bf Proof of Theorem \hopfrinow{}.} \ The 
implications $(iii)\Rightarrow(ii)\Rightarrow(i)\Rightarrow(iv)$
are trivial. To see that $(iv)\Rightarrow(iii)$, we assume that $(iv)$
holds, pick an $\bar x\in M$, and $r>0$, and a sequence 
$\{x_j\}$ in $B_{{\cal M}}(\bar x,r)$, and try to find a
convergent subsequence and a length minimizer joining $\bar x$ to the
limit of this subsequence. 

Pick arcs $\xi_j\in{\cal{}A}^{PAL,0}({\cal M})$ such that
$\partial\xi_j=(\bar x,x_j)$ and 
\xdef\difiu{A.\the\equano}
$$\
L_j\,\,\equaldef\,\,||\xi_j||_{_G}\le d_{{\cal M}}(\bar x,x_j)+\frac{1}{j}\,.\numeqa
$$
Then $L_j\le r+1$ for all $j$. Let $I(\xi_j)=[0,L_j]$.   Since the
sequence $\{L_j\}$ is bounded we may assume, after passing to a
subsequence,  that $L=\lim_{j\to\infty}L_j$ exists. 

Let $\tilde\xi_j\in\maxi{{\cal C}^{a,\le 1}({\cal{}M})}$ be an extension of
$\xi_j$. (The existence of $\tilde\xi_j$ follows easily from Zorn's
Lemma.)  Since $(iv)$ holds, the domain $I(\tilde\xi_j)$ of each
$\tilde\xi_j$ is the whole real line. By Proposition \compaprop{}, there
exist an infinite subset $S$ of $\nn$ and a curve 
$\tilde\xi\in\maxi{{\cal{}C}^{a,\le 1}({\cal{}M})}$ 
such that $0\in I(\tilde\xi)$ and 
$\tilde\xi_j\to\tilde\xi$ on compact intervals. Since $(iii)$ holds,
$I(\tilde\xi)$ is in fact equal to $\rr$. In particular, this implies
that the sequence $\{\xi_j(L_j)\}_{j\in S}$ converges to
$\tilde\xi(L)$, so the limit $x=\lim_{j\to_S\infty}x_j$ exists and
equals $\tilde\xi(L)$.

Let $\xi$ be the restriction of $\tilde\xi$ to $[0,L]$.
If $t_1,t_2\in[0,L]$, then $\xi(t_\ell)=\lim_{j\to_{_S}\infty}\xi_j(t_\ell)$
for $\ell=1,2$. So
$d_{{\cal M}}(\xi(t_1),\xi(t_2))=\lim_{j\to_{_S}\infty}
d_{{\cal{}M}}(\tilde{\xi}_j(t_1),\tilde{\xi}_j(t_2))$. 
Since $\tilde{\xi}_j\in{\cal C}^{a,\le 1}({\cal{}M})$, 
Proposition \proplipimplies{} implies that
$d_{{\cal{}M}}(\tilde{\xi}_j(t_1),\tilde{\xi}_j(t_2))\le|t-t'|$. So
$d_{{\cal{}M}}(\xi(t_1),\xi(t_2)\le|t_1-t_2|$ for all $t_1,t_2\in[0,L]$. 
Then Proposition \proplipimplies{} tells us that
$\xi\in{\cal C}^{a,\le 1}({\cal{}M})$. This implies, in particular,
that $||\xi||_{_G}\le L$, and $d_{{\cal M}}(\bar x,x)\le||\xi||_{_G}$. On
the other hand, 
$$
L=\lim_{j\to\infty}L_j\le\lim_{j\to_S\infty}
d_{{\cal{}M}}(\bar{x},x_j)+\frac{1}{j}=
d_{{\cal{}M}}(\bar{x},x)\,.
$$
So $||\xi||_{_G}\le d_{{\cal{}M}}(\bar{x},x)$, and then 
$||\xi||_{_G}=d_{{\cal{}M}}(\bar{x},x)$.  Therefore 
$\xi$ is a length minimizer, and the implication 
$(iv)\Rightarrow(iii)$ is proved.

Now assume that $LMin^{PAL}({\cal M})$ has the extension property.
We  prove that $(i)\Rightarrow(v)$. Let 
$\xi\in\maxi{LMin^{PAL}({\cal M})}$. Then the extension property implies,
to begin with, that $I(\xi)$ is open, and in particular nontrivial.
Let $b=\sup I(\xi)$, $a=\inf I(\xi)$. Then
$a\notin I(\xi)$ and $b\notin I(\xi)$.  If $b$ was finite, then the
completeness of $(M,d_{{\cal M}})$ would imply that
$\lim_{t\uparrow{}b}\xi(t)$ exists, since
$d_{{\cal{}M}}(\xi(t),\xi(t')) \le|t-t'|$ whenever $t,t'\in I(\xi)$,
because $\xi$ is parametrized by arc-length.  But then the extension
property would imply that $b\in{\rm Int}(I(\xi))$, which
is a contradiction. So $b=+\infty$. Similarly, $a=-\infty$.

We now prove that $(v)\Longrightarrow(iii)$.  We pick $\bar x\in M$,
$r>0$, and a sequence $\{x_j\}$ in $B_{{\cal M}}(\bar x,r)$, and
try to find a convergent subsequence and a length minimizer joining
$\bar x$ to the limit $\hat x$ of this subsequence. 
Pick arcs $\xi_j\in{\cal{}A}^{PAL,0}({\cal M})$ such that
$\partial\xi_j=(\bar x,x_j)$ and (\difiu) holds. 
Then $L_j\le r+1$ for all $j$. Let $I(\xi_j)=[0,L_j]$.   Since the
sequence $\{L_j\}$ is bounded we may assume, after passing to a
subsequence,  that $L=\lim_{j\to\infty}L_j$ exists. 

Let $\tilde\xi_j\in\maxi{{\cal C}^{a,\le 1}({\cal{}M})}$ be an extension of
$\xi_j$. (The existence of $\tilde\xi_j$ follows easily from Zorn's Lemma.)  By
Proposition \compaprop{}, there exist an infinite subset $S$ of $\nn$ and a
curve $\tilde\xi\in\maxi{{\cal{}C}^{a,\le 1}({\cal{}M})}$ such that $0\in
I(\tilde\xi)$ and $\{\tilde\xi_j\}_{j\in S}$ converges to $\tilde\xi$ on
compact intervals. It is then clear that $\tilde\xi(0)=\bar x$.

We claim that $L\in I(\tilde\xi)$.  To see this, we
first observe that, since $\tilde\xi\in\maxi{{\cal C}^{a,\le
1}({\cal{}M})}$, the extension property implies that $I(\tilde\xi)$ is
open. Let $b=\sup\,I(\tilde\xi)$, so $b>0$, because \hbox{$0\in I(\tilde\xi)$.}
Let $\tau$ be such that $0<\tau\le L$ and $\tau<b$. Then the compact
interval $[0,\tau]$ is contained in $I(\tilde\xi)$. So 
$[0,\tau]\subseteq I(\tilde\xi_j)$ if $j\in S$ is
sufficiently large, and
\hbox{$\tilde\xi_j\restr[0,\tau]\to\tilde\xi\restr[0,\tau]$}
\hbox{uniformly as $j\to_{_S}\infty$.} If $t,t'$ belong to the interval 
$[0,\tau]$, then 
\hbox{$\tilde\xi(t)=\lim_{j\to_{_S}\infty}\xi_j(t)$,}
and \hbox{$\tilde\xi(t')=\lim_{j\to_{_S}\infty}\xi_j(t')$, so 
$d_{{\cal M}}(\tilde\xi(t'),\tilde\xi(t))=\lim_{j\to_{_S}\infty}
d_{{\cal{}M}}(\xi_j(t'),\xi_j(t))$.}  Since the arc $\xi_j$ is
\hbox{parametrized by arc-length, we have}
$d_{{\cal{}M}}(\xi_j(t'),\xi_j(t))\le|t'-t|$, 
so 
\xdef\islesseq{A.\the\equano}
$$
d_{{\cal M}}(\tilde\xi(t'),\tilde\xi(t))\le|t'-t|\,.\numeqa
$$
On the other hand (assuming, say, that $t<t'$), the triangle
inequality yields
$$
d_{{\cal M}}(\xi_j(0),\xi_j(t))+
d_{{\cal M}}(\xi_j(t),\xi_j(t'))+d_{{\cal M}}(\xi_j(t'),\xi_j(L_j))\ge
d_{{\cal M}}(\xi_j(0),\xi_j(L_j))\ge L_j-{1\over j}\,.
$$
Since $d_{{\cal M}}(\xi_j(0),\xi_j(t))\le||\xi_j\restr[0,t]||_{_G}=t$
and 
$d_{{\cal M}}(\xi_j(t'),\xi_j(L_j))\le||\xi_j\restr[t',L_j]||_{_G}=L_j-t'$, 
it follows that $d_{{\cal M}}(\xi_j(t),\xi_j(t'))\ge|t'-t|-{1\over j}$.
If we then let $j\to_{_S}\infty$, we find that the inequality 
$d_{{\cal M}}(\tilde\xi(t),\tilde\xi(t'))\ge|t'-t|$ holds.
This, together with (\islesseq), and the fact that $\tau$ was any 
number such that $0\le \tau\le L$ and $\tau<b$,  yields
\xdef\islocmin{A.\the\equano}
$$
d_{{\cal M}}(\tilde\xi(t),\tilde\xi(t'))=|t'-t|\quad{\rm
  whenever}\quad
t,t'\in[0,L]\cap[0,b\,[\,\,.\numeqa
$$
Now let $\xi$ be the restriction of $\tilde\xi$ to the interval 
$[0,L]\cap[0,b\,[\,$.  Then (\islocmin) implies that $\xi$ is a local
minimizer parametrized by arc-length. Let $\hat\xi$ be an extension of 
$\xi$ such that $\hat\xi\in\maxi{LMin^{PAL}}$.  (The
existence of $\hat\xi$ follows from Zorn's lemma.) Then 
the fact that $LMin^{PAL}$ has the extension property implies that 
$I(\hat\xi)=\rr$. 

If $b\le L$, then $\hat\xi(b)=\lim_{t\uparrow b}\xi(t)$,
so $\lim_{j\to_{_S}\infty}\tilde\xi(t)=\lim_{j\to_{_S}\infty}\xi(t)$
exists. Then the extension property for ${\cal C}^{a,\le  1}({\cal{}M})$
implies that $b\in{\rm Int}(I(\tilde\xi))$, contradicting the fact 
that $b=\sup I(\tilde\xi)$.  
So $b>L$. But then (\islocmin) says that
\xdef\islocmin{A.\the\equano}
$$
d_{{\cal M}}(\xi(t),\xi(t'))=|t'-t|\quad{\rm  whenever}\quad
t,t'\in[0,L]\,.\numeqa
$$
Let $\hat x=\xi(L)$. Then $\xi_j(L)\to\hat x$ as $j\to_{_S}\infty$,
and 
$$
d_{{\cal M}}(\xi_j(L),x_j)= d_{{\cal
    M}}(\xi_j(L),\xi_j(L_j))\le|L-L_j|\to 0\,,
$$
so $x_j\to\hat x$  as $j\to_{_S}\infty$. Moreover, 
since $\xi(0)=\bar x$, (\islocmin) implies that 
$d_{{\cal M}}(\bar x,\hat x)=L$. Since $\partial\xi=(\bar x,\hat x)$,
and $||\xi||_{_G}\le L$, we see that $\xi$ is a length minimizer joining
$\bar x$ to $\hat x$. The proof is then complete. \proofend


\medbreak\noindent{\bf Proof of Proposition \eximinloc{}.}\ Let 
$\rho=d_{{\cal M}}(\bar x,\hat x)$, and
pick a sequence $\{\xi_j\}$ in ${\cal A}^{PAL,0}({\cal M})$
such that $\partial\xi_j=(\bar x,\hat x)$ and $\tau_+\xi_j\le\rho+\frac{1}{j}$.
Then $\rho\le\tau_+\xi_j$.  Let $\tilde\xi_j$ denote 
the restriction of
$\xi_j$ to $[0,\rho]$.  Then $\tilde\xi_j$ is contained in 
$B_{{\cal{}M}}(\bar x,r)$.

Since the $\tilde\xi_j$ are parametrized by arc-length, the inequality 
$d_{{\cal M}}(\tilde\xi_j(t'),\tilde\xi_j(t))\le|t'-t|$
holds for all $t,t'\in[0,\rho]$. Since all the $\tilde\xi_j$ are contained in 
$B_{{\cal{}M}}(\bar x,r)$, which is compact, the Ascoli-Arzel\`a theorem
implies that the sequence $\{\tilde\xi_j\}$ 
has a subsequence $\{\tilde\xi_{j(k)}\}$ that converges uniformly to 
an arc $\xi:[0,\rho]\to M$.   Then 
$d_{{\cal M}}(\xi(t'),\xi_j(t))\le|t'-t|$ for all
$t,t'\in[0,\rho]$. So Proposition \proplipimplies{} tells us that 
$\xi$ is an $E$-admissible $E$-arc 
such that $||\dot\xi(t)||_{_G}\le 1$ for almost all $t$.  
On the other hand,  $d_{{\cal M}}(\hat x,\tilde\xi_{j(k)}(\rho))\le
\frac{1}{j}$, so  $\tilde\xi_{j(k)}(\rho)\to \hat x$. Therefore
$\xi(\rho)=\hat x$. Since $\rho=d_{{\cal M}}(\bar x,\hat x)$, \
and $||\dot\xi(t)||_{_G}\le 1$ for almost all $t\in[0,\rho]$,
we must have $||\xi||_{_G}=\rho$, so $\xi$ is a length minimizer.  \proofend
  
\medbreak\noindent{\bf Proof of Statement ($*$) of \S\secextre{}.} \ We 
work locally, in a 
neighborhood $U$ of $x$ where there is a coordinate chart $\bx$ and an
orthonormal basis ${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$.  Then
the section $V$ is  given by 
$V(x',p')=\sum_{i=1}^mV^i(x',p')f_i(x')$,
where the $V^i$ are smooth functions in a neighborhood of $(x,p)$. 
If $W$ is another section of $\hat F$ such that $W(x,p)=v$, then 
$W$ also has an expression
$W(x',p')=\sum_{i=1}^mW^i(x',p')f_i(x')$,
and the functions $W^i$ satisfy $W^i(x,p)=V^i(x,p)$ for
$i=1,\ldots,m$. Then 
$H_{{\cal M}}^{c,W,p_0}-H_{{\cal M}}^{c,V,p_0}$ vanishes at $(x,p)$, 
so 
$$
H_{{\cal M}}^{c,W,p_0}(x',p')-H_{{\cal M}}^{c,V,p_0}(x',p')
=\sum_{i=1}^m\Big(W^i(x',p')-V^i(x',p')\Big)h_i(x',p')
$$
near $(x,p)$, where the $h^i$ are smooth functions.  Therefore
\xdef\says{A.\the\equano}
$$
\vec{H}_{{\cal M}}^{c,W,p_0}(x,p)-\vec{H}_{{\cal M}}^{c,V,p_0}(x',p')
=\sum_{i=1}^m\Big(W^i(x,p)-V^i(x,p)\Big)\vec{h}_i(x,p)
+\sum_{i=1}^mh_i(x,p)z^i\,,\numeqa
$$
where $z^i=\vec{Z}^i(x,p)$, and $Z^i=W^i-V^i$.  

The fact that $(x,p,v,p_0)$ belongs to $\,Crit({\cal M})$ implies that
$h_i(x,p)=0$. Since $W^i(x,p)-V^i(x,p)=0$ as well, (\says) 
says that 
$$
\vec{H}_{{\cal M}}^{c,W,p_0}(x,p)=\vec{H}_{{\cal M}}^{c,V,p_0}(x,p)\,,
$$
and the proof of ($*$) is complete. \proofend

\medbreak\noindent{\bf Proof of Proposition \efintrinsic{}.} \ The 
fact that $(\xi(t),\psi(t),\omega_{\xi}(t),\psi_0)\in Crit({\cal M})$ is an
immediate consequence of (E.2).   Fix a $\bar t\in I$, and let $\bar
x=\xi(\bar t)$. Let  $U$ be a neighborhood of $\bar x$ which is contained 
in the domain $U_{{\bf f}}$ of a local orthonormal basis 
${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$ and also in the domain 
${\rm Dom}(\bx)$ of a chart $\bx$ near $\bar x$.  Let 
$g_i=\sigma(f_i)$.   Fix a $t\in I$ such that
$\xi(t)\in U$.
Write $\omega_{\xi}(t)=\sum_{i=1}^mv^if_i(\xi(t))$, and define 
$V(x)=\sum_{i=1}^mv^if_i(x)$, for $x\in U$. Then the Hamiltonian function
$H_{{\cal M}}^{c,V,\psi_0}$ is given, near $(\xi(t),\psi(t))$, by 
$$
H_{{\cal M}}^{c,V,\psi_0}(x,p)=
\sum_{i=1}^mv^i\mylangle p,g_i(x)\myrangle-{\psi_0\over 2}\sum_{i=1}^m(v^i)^2\,.
$$
This implies that 
$$
\eqalign{
\vec{H}_{{\cal{}M}}^c(\Xi(t),\omega_{\xi}(t),\psi_0)&=
\vec{H}_{{\cal M}}^{c,V,\psi_0}(\xi(t),\psi(t))\cr
&=\sum_{i=1}^mv^i.g_i^j(\xi(t)).{\partial\,\,\,\over\partial x^j}
-\sum_{i=1}^mv^i.{\partial \mu_{_{g_i}}\over\partial x^j\,\,}
\Big(\xi(t),\psi(t)\Big).{\partial\,\,\,\over\partial p_j}\,,\cr}
$$
where $\mu_{_{g_i}}$ is the momentum function corresponding to $g_i$
---defined by Formula (\flamomen)--- 
and we have written $g_i=\sum_{i=1}^mg_i^j.{\partial\,\,\,\over\partial
x^j}$. 
So $\vec{H}_{{\cal M}}^{c,V,\psi_0}(\xi(t),\psi(t))$ is exactly equal to
$\dot\Xi(t)$, and our proof is complete. \proofend


\medbreak\noindent{\bf Proof of Theorem \minisareopt{}.} \ We 
let ${\cal U}$ denote the set $\Gamma(F)$ of all smooth sections of $F$.
We define 
$$
w(x,u)=\sigma(u)(x)\,,\quad L(x,u)={1\over 2}||u(x)||_{_G}^2\quad{\rm for}\quad
u\in{\cal U}\,,  x\in M\,.
$$
\def\pleft#1{\hbox to2.2cm{#1\hfill}}
\def\pleftb#1{\hbox to4cm{#1\hfill}}
We then consider the optimal control problem 
$$
(P)\,\,\,\,\,\,\,\,\left\{\matrix{\pleft{{\rm minimize}}&
\pleftb{$\int_a^bL(\xi(t),\eta(t))\,dt$}\cr
\pleft{subject to}&
\pleftb{$\dot\xi(t)\,=\,w(\xi(t),\eta(t))\,\,{\rm a.e.}$}\cr 
\pleft{{\rm and}}&\pleftb{$\,\,\,\,\partial\xi\,=\,\partial\xi_*\,\,,$}\cr}\right.
$$
where $\xi$ ranges over the space of absolutely continuous $M$-valued
curves on $[a,b]$, and $\eta$ takes values in the class 
$Adm_{[a,b]}({\cal U})$ of ``admissible controls on $[a,b]$,'' that is, the
set of all maps $[a,b]\ni t\to\eta(t)\in{\cal U}$ such that 
(i)~the map $t\to w(x,\eta(t))$ is measurable for each $x\in M$, and 
(ii)~for each chart $\bx$ and each compact subset $K$ of ${\rm Dom}(\bx)$
there exists an integrable function $\ffi:[a,b]\to[0,\infty]$ with the
property that 
$$
||w^{\bx}(x,\eta(t))||+\Big\|{\partial w^{\bx}\over\partial x\,\,}
(x,\eta(t))\Big\|\le\ffi(t)\quad{\rm whenever}\quad x\in K,t\in[a,b]\,,
$$
where $w^{\bx}$ is the vector-valued function that expresses $w$ 
in the chart $\bx$.   

Let $[a,b]\ni t\to\eta_*(t)\in{\cal U}$ be an admissible control such that
$\eta_*(t)(\xi_*(t))=\omega_{\xi_*}(t)$ for almost all $t$.  (The existence
of $\eta_*$ is easily proved: it suffices to work locally, i.~e. to assume
that $\xi_*$ is contained in the domain $U_{{\bf f}}$ of an orthonormal
basis ${\bf f}=(f_1,\ldots,f_m)$ of sections of $F$; we then write
$\omega_{\xi_*}(t)=\sum_{i=1}^m v^i(t)f_i(\xi_*(t))$, where the $v^i$ are
integrable functions; we let $\zeta$ be a smooth compactly supported
function on $M$ 
which is equal to $1$ on a neighborhood of $\xi_*([a,b])$ and whose support
is contained in $U_{{\bf f}}$; we then define the smooth section
$\eta_*(t)$ of $F$ by letting
$\eta_*(t)(x)=\sum_{i=1}^mv^i(t).\zeta(x).f_i(x)$.)

It is then clear that the pair $(\xi_*,\eta_*)$ is a solution of (P). So
the Pontryagin principle implies that there exist an absolutely continuous
field of covectors $t\to\psi(t)$ along $\xi$ and a real number $\psi_0$
such that (E.1) holds, as well as the Hamiltonian maximization
condition (HMC) and the adjoint equation. The HMC says that, if we let 
$$
h(x,p,u,p_0)\equaldef\mylangle p,w(x,u)\myrangle-p_0L(x,u)
$$
for $(x,p)\in T^*M$, $p_0\in\rr$, $u\in{\cal U}$, then for almost all $t$
the function 
$$
{\cal U}\ni u\to h(\xi_*(t),\psi(t),u,\psi_0)
$$
is maximized by $u=\eta_*(t)$.  Since $h(x,p,u,p_0)$ only depends, for a
given $(x,p,p_0)$, on the value $u(x)$ of $u$ at $x$, the HMC for a given
$t$ is equivalent to the statement that the function $F(\xi_*(t))\ni
y\to\mylangle\psi(t),\sigma_{\xi_*(t)}(y)\myrangle- {\psi_0\over 2}||y||_{_G}^2$
is maximized by $y=\eta_*(t)(\xi_*(t))$, i.~e. by $y=\omega_{\xi_*}(t)$. So
the HMC is exactly equivalent to (E.2).  It then follows from Proposition
\efintrinsic{} that $(\xi_*(t),\psi(t),\omega_{\xi_*}(t),\psi_0)\in
Crit({\cal M})$ for almost every $t\in[a,b]$.

Finally, the adjoint equation says, relative to a chart $\bx$, that 
\xdef\duvalla{A.\the\equano}
$$
\dot\psi(t)=-{\partial h\over\partial x}(\xi_*(t),\psi(t),\eta_*(t),\psi_0)
\quad {\rm for\,\,\,a.~e.}\quad t\,.\numeqa
$$
If we define 
$$
h_{u,p_0}(x,p)\,\equaldef\,h(x,p,u,p_0)\,,
$$
and let $\Xi_*(t)=(\xi_*(t),\psi(t))$, then \duvalla{} implies that
\xdef\duvallb{A.\the\equano}
$$
\dot\psi(t)=-{\partial h_{\eta_*(t),\psi_0}\over\partial x
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}
(\Xi_*(t))\quad {\rm for\,\,\,a.~e.}\quad t\,.\numeqa
$$
In addition, it is clear that 
${\partial h_{u,p_0}\over\partial p\,\,\,\,}(x,p)=w(x,u)$, so 
\xdef\duvallc{A.\the\equano}
$$
\dot\xi_*(t)={\partial h_{\eta_*(t),\psi_0}\over\partial p
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}
(\Xi_*(t))\quad {\rm for\,\,\,a.~e.}\quad t\,.\numeqa
$$
It follows from (\duvallb) and  (\duvallc) that 
\xdef\duvalld{A.\the\equano}
$$
\dot\Xi_*(t)=\vec{h}_{\eta_*(t),\psi_0}(\Xi_*(t))\quad {\rm
for\,\,\,a.~e.}\quad t\,.\numeqa
$$
Now, if we write $\eta_*(t)(x)=\sum_{i=1}^m\theta^i(x,t)f_i(x)$, and 
define $V_t(x)=\sum_{i=1}^m\theta^i(x,t)f_i(x)$, then $V_t$ is, for each
$t$, a smooth section of $F$ near $\xi_*(t)$, such that 
$V_t(\xi_*(t))=\omega_{\xi_*}(t)$, and
\xdef\duvalle{A.\the\equano}
$$
H_{{\cal M}}^{c,V_t,\psi_0}(x,p)=h(x,p,\eta_*(t),\psi_0)=
h_{\eta_*(t),\psi_0}(x,p)
\quad{\rm for}\quad(x,p)\,\,\,{\rm near}\,\,\,(\xi_*(t),\psi(t))\,.\numeqa
$$
Then (\duvalld) and  (\duvalle) imply
\xdef\duvallf{A.\the\equano}
$$
\dot\Xi_*(t)=\vec{H}_{{\cal M}}^{c,V_t,\psi_0}(\Xi_*(t))\quad {\rm
for\,\,\,a.~e.}\quad t\,.\numeqa
$$
Since $(\xi_*(t),\psi(t),\omega_{\xi_*}(t),\psi_0)\in Crit({\cal M})$ for
almost every $t\in[a,b]$, and $V_t$ is, for each $t$, a smooth section of
$F$ near $\xi_*(t)$ such that $V_t(\xi_*(t))=\omega_{\xi_*}(t)$, the
definition of  $\vec{H}_{{\cal M}}^c$ implies that 
\xdef\duvallg{A.\the\equano}
$$
\vec{H}_{{\cal M}}^{c,V_t,\psi_0}(\Xi_*(t))
\,\,=\,\,\vec{H}_{{\cal M}}^{c}(\Xi_*(t),\omega_{\xi_*}(t),\psi_0)\quad {\rm
for\,\,\,a.~e.}\quad t\,.\numeqa
$$
Therefore 
\xdef\duvallh{A.\the\equano}
$$
\dot\Xi_*(t)=\vec{H}_{{\cal M}}^{c}(\Xi_*(t),\omega_{\xi_*}(t),\psi_0)\quad
{\rm for\,\,\,a.~e.}\quad t\,.\numeqa 
$$
Then Proposition \efintrinsic{} implies that (E.3) holds, and $\xi_*$ is an
extremal. \proofend

\medbreak\noindent{\bf Proof of Proposition \iscanonical{}.}\ Let 
$\eta^i(t)=\tilde\ffi_{{\bf f}}^i(\xi(t),\psi(t))$,
$\eta(t)=(\eta^1(t),\ldots,\eta^m(t))$.  We have to prove 
that $||\eta(t)||=||\dot\xi(t)||_{_G}$ for a.~e. $t$.  To see this, fix
a $t\in I(\xi)$  such that $\dot\xi(t)=\sum_{i=1}^m\eta^i(t)g_i(\xi(t))$. 
Then for any other representation 
$\dot\xi(t)=\sum_{i=1}^m(\eta^i(t)+\nu^i)g_i(\xi(t))$ 
of $\dot\xi(t)$ as a linear combination
of the vectors $g_i(\xi(t))$, the sum $\sum_{i=1}^m\nu^i.g_i(\xi(t))$ 
vanishes. So 
$\sum_{i=1}^m\nu^i.\eta^i(t)=\sum_{i=1}^m\nu^i.\mylangle
p,g_i(\xi(t))\myrangle=\mylangle p,\sum_{i=1}^m\nu^i.g_i(\xi(t))\myrangle=0$. 
It then follows that 
$||\eta(t)+\nu||^2=||\eta(t)||^2+||\nu||^2\ge ||\eta(t)||^2$.
So $\eta(t)=\eta_{{\bf f},\xi,I(\xi)}(t)$. \proofend

\medbreak\noindent{\bf Proof of Proposition \sbgoldchar{}.} \ Let 
$p\in T_x^*M\backslash\{0\}$ be such that 
$H_{{\cal M}}(x,p)=0$. If $v,w$ are any two vectors in $F(x)$
and $v=\sum_iv^if_i(x)$, $w=\sum_iw^if_i(x)$, then 
an easy calculation shows that 
$Z(x,p)(v,w)=\sum_{i,j}\zeta_{i,j}(x,p)v^iw^j$, 
where $\zeta_{i,j}(x,p)=\mylangle p,[g_i,g_j](x)\myrangle$.
Let $v\in F(x)$ be such that $v\neq 0$. Let 
$\theta=(v^1,\ldots,v^m)$. Then 
\xdef\ispsieq{A.\the\equano}
$$
Z(x,p)(v,w)=0\quad{\rm for\,\,\, all}\quad w\in F(x)\numeqa
$$
if and only if 
$\sum_{i}\mylangle p,[v^ig_i,g_j](x)\myrangle=0$ for all $j$, i.~e.
iff $\mylangle p,[{\bf g}\theta,g_j](x)\myrangle=0$ for all $j$ or,
equivalently, iff 
$p$ is orthogonal to all the vectors $[{\bf g}\theta,g_j](x)$, for
$j=1,\ldots,m$. Since $p$ is also orthogonal to the vectors $g_j(x)$, for
$j=1,\ldots,m$, because  $H_{{\cal M}}(x,p)=0$, it follows that 
(\ispsieq) is equivalent to 
\xdef\ispsieqb{A.\the\equano}
$$
p\quad{\rm is\,\,\, orthogonal\,\,\, to\,\,\,the\,\,\,linear\,\,\, span
\,\,\,of\,\,\,the}\quad g_j(x)\quad{\rm and\,\,\,the}\quad
[{\bf g}\theta,g_j](x)\,.\numeqa
$$
If $E$ is SBG at $x$, then a nonzero $p$ and a nonzero $\theta$ 
for which (\ispsieqb) holds cannot exist, for any such 
pair $(p,\theta)$ would give rise to a nonzero vector $v\in F(x)$ 
belonging to the kernel of the bilinear form $Z(x,p)$, and this
would contradict the SBG condition, since (\ispsieqb) implies that 
$H_{{\cal M}}(x,p)=0$, so $Z(x,p)$ is nonsingular. So, if 
$\theta\neq 0$, then the span of 
the vectors $g_j(x)$  and $[{\bf g}\theta,g_j](x)$ must be the whole
space $T_xM$, showing that (SBG$_{loc}$) holds. 

Conversely, if (SBG$_{loc}$) holds, then for every nonzero $\theta$ the
span of the vectors $g_j(x)$ and $[{\bf g}\theta,g_j](x)$ is the whole
space $T_xM$. Therefore there cannot exist 
a nonzero $p$ and a nonzero $\theta$ for which (\ispsieqb) holds. 
So, if $p\neq 0$ is such that $H_{{\cal M}}(x,p)=0$, and $v\neq 0$, 
then (\ispsieq) cannot hold either. Therefore $Z(x,p)$ is
nonsingular, so $E$ is SBG at $x$. \proofend




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