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\title[Curvature Tensors and Representations]{The Representation Theory of
  \\
 Riemannian Curvature Tensors}    
\author{Roe Goodman}      
\institute{CUNY Graduate Center -- Representation Theory Seminar}    
\date{February 19, 2010}       % Enter the date or \today between curly braces


%%%%%%%%%%%%%  end of preamble

\begin{document}

\begin{frame}
  \titlepage
\end{frame}
\note{ } % Add notes to yourself that will be displayed when
                           % typeset with the notes class option

% \section{Outline}

% Creates table of contents slide incorporating
% all \section and \subsection commands
\begin{frame}
\frametitle{Outline}
  \tableofcontents[pausesections]

\pause
H. Weyl:
\ \alert{Reine Infinitesimalgeometrie}  (1918)

\quad \alert{Das gruppentheoretische Fundament der Tensorrechnung} (1924) 

\pause

R. S. Kulkani:
\ \alert{On the Bianchi Identities} (1972) 

A. Besse:
\ \alert{Einstein Manifolds} (1987) 

R. S. Strichartz:
\ \alert{Linear Algebra of Curvature Tensors 

\quad  and their Covariant Derivatives} (1988)

\end{frame}

\section{Riemannian Connection and Curvature Tensor}

%%%%%%%%%%%   frame 1 %%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Riemannian Connection and Curvature Tensor} 

    $(M, g)$ -- smooth (pseudo) Riemannian manifold: 

   nondegenerate bilinear form $g_p$ on tangent space $T_p(M)$

   (Riemannian if $g_p$ positive definite)
%
\pause    

    $\cC(M)$ -- smooth functions \quad
    $\cT(M)$ -- smooth vector fields

\pause 
%  
   Riemannian connection: 
   
 $X \in \cT(M)$ acts as \alert{covariant derivative} $\nabla_X$ on tensor fields: 

  \begin{itemize}
%  
  \item  $\nabla_{\varphi X}Y = \varphi\nabla_X Y$ 
 \qquad $\nabla_X(\varphi Y) = X(\varphi)Y  + \nabla_X Y$

\end{itemize}

for $\varphi \in \cC(M)$ and $Y \in \cT(M)$ 

\pause

  $\nabla$ uniquely determined from $g$ by requiring
  
    \begin{itemize}
%
    \item
      \alert{covariant constant metric tensor}: 
$  X(g(Y,Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z) $
%
\pause
    \item   
     \alert{zero torsion}: \quad
%
 $     \nabla_X Y - \nabla_Y X = [X, Y]  $
%
    \end{itemize}
%

\end{frame}

%%%%%%%%%%%%%%%%%%% frame 2 %%%%%%%%%%%%%%%%
\begin{frame}

\alert{Curvature tensor field} $R_p(x,y) \in \End(T_p M) = T_p M   \otimes (T_p M)^*$ :

\vspace{.5ex}

\quad $  R_p(x,y)z = \left(\nabla_{X}\nabla_{Y}Z - \nabla_{Y}\nabla_{X}Z 
     - \nabla_{[X,Y]}Z\right)_{\!p}  $

\pause
\begin{itemize}

\item Only depends on $X_p = x$, $Y_p = y$, and $Z_p = z$ (\alert{tensorial})
(unlike \alert{Lie derivative} $\theta(X)$ --  not
tensorial)
\pause

\item Measures failure of $X \mapsto \nabla_X$ to be Lie algebra homomorphism
 

\end{itemize}
\pause

\alert{Algebraic Symmetries} of the curvature tensor:
 
 \begin{itemize}  
%  
  \item[(C1)]  
    $ R_p(x,y) = - R_p(y,x)$, so  $R_p: \bigwedge^2 T_p(M) \to \End(T_p(M))$
%
\pause
%
  \item[(C2)]
      $ R_p(x, y)^{*}  = - R_p(x, y)$ ($*$ via $g_p$), so  $R_p(x,y) \in
        \Lie({\rm O}(g_p))$
%
\pause
%
  \item[(C3)] Jacobi identity for $\cT(M)$ + zero torsion $\Longrightarrow$ 
   \alert{Bianchi identity}:
 $   R_p(x,y)z +   R_p(y,z)x +  R_p(z,x)y = 0 $
%
 \end{itemize}
%

\end{frame}
\note[enumerate]       % Add notes to yourself that will be displayed when
{                      % typeset with the notes class option
\item Note for Point 1   
\item Note for Point 2   
}

\section{The Space of Curvature Tensors}

%%%%%%%%%%%%%%%%%%%% frame 3 %%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Space of Curvature Tensors}

Fix $p \in M$. Let $E = (T_pM)_{\bC} \cong E^*$ \quad (via $Q = (g_p)_{\bC}$)

Define \alert{Riemann-Christoffel curvature tensor} $R \in \bigotimes^4 E
\cong \bigotimes^4 E^{*}$:
%
\[
  R(v, w, x, y) =  Q(R_p(v, w)x, y )
\quad\mbox{for $v,w,x,y \in E$}  
\]
%
\pause
%
\alert{Program:} Study subspace $\Curv(E) \subset \bigotimes^4 E$ 
of tensors with algebraic symmetries (C1) + (C2) + (C3).

\pause
Let $R \in \Curv(E)$, 
$\sigma = $ permutation representation of $\frS_4$ on $\bigotimes^4 E$ 

\pause

\begin{itemize}
\item (C1) + (C2) + (C3) \ $\Longrightarrow$ \ 
%
$\sigma(12)R = -R$, \quad $\sigma(34)R = -R$,
%
$\sigma(13)\sigma(24)R = R $
%
\quad $\Longrightarrow$ \quad  $R \in S^2(\bigwedge^2 E)$
%
%
\pause
\item $S^2(\bigwedge^2 E) $ is invariant under the \alert{Bianchi operator} 
%
$b = \rule{0ex}{2.5ex} \frac{1}{3}(I + \sigma(123) + \sigma(123)^2) =
 \frac{1}{3}(I + \sigma(13)\sigma(12) + \sigma(23)\sigma(12))$
%
\end{itemize}
%
\pause

Hence $\Curv(E) = \Ker(b) \cap S^2(\bigwedge^2 E)$

\end{frame}

%%%%%%%%%%%%%%%%%%%%% frame 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{$\Curv(E)$ as $\GL(E)$ module} 

\begin{itemize}
\item
$\Range(b) \cap S^2(\bigwedge^2 E) = \bigwedge^4 E$ 
\quad  (irreducible for $\GL(E)$)

\pause
\item
 $b^2 = b$ $\Longrightarrow$ 
 $ S^2(\bigwedge^2 E) =  \Curv(E) \oplus \, \big(\bigwedge^4 E \big) $ 
 
(second summand zero if $\dim E < 4$)
\pause
\item
$ \dim \Curv(\bC^n) = \frac{1}{12} \,n^2(n+1)(n - 1)$ \quad\alert{($\star$)}
\pause
\end{itemize}

Fix $Q$-orthonormal basis $e_1, \ldots, e_n$ for $E \cong \bC^n$ ($n \geq 2)$

Let $\lambda = [\lambda_1, \ldots, \lambda_k] \in \bN^k$, 
\quad $\lambda_1 \geq \cdots \geq  \lambda_k > 0$ 
 \quad ($k \leq n$) 

$F^{\lambda}_n =$ irreducible $\GL(n, \bC)$ representation, 
  highest weight $\lambda$

\pause

\begin{theorem}
%
$\Curv(\bC^n) \cong F_n^{[2,2]}$ is an irreducible $\GL(n,\bC)$ module.
%
\end{theorem}

\pause
\begin{Proof}

High wt vector  $R = (e_1\wedge e_2)\otimes(e_1\wedge e_2)$
 $\Longrightarrow$ $F^{[2, 2]}_n \subset \Curv(\bC^n)$.

\pause
Weyl dim. formula +  \alert{($\star$)} $\Longrightarrow$ 
$\dim F^{[2, 2]}_n = \dim \Curv(\bC^n)$
\end{Proof}

\end{frame}


%%%%%%%%%%%%%%%%%%%%% frame 5 %%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Curvature and Young Symmetrizers}
 
Young tableau $A$ ($k$ boxes) has  \alert{row group} and \alert{column
group}

\pause
\alert{Young symmetrizer:} \quad $\bp_A \in \End_{\GL(n,\bC)}(\bigotimes^k \bC^n)$

\quad  (alternate over column group)$\cdot$(symmetrize over row group)

\pause

\alert{Weyl module}: $\Range\bp_A$ is an irreducible $\GL(n, \bC)$ module,

\quad highest weight $\lambda = \shape(A)$

\pause
\begin{Corollary}
$\Curv(\bC^n) = \Range \bp_A$ \ where \ $A=\begin{array}{l} 
 \fbox{$1$}\fbox{$3$} 
  \\
  \fbox{$2$}\fbox{$4$} 
\end{array}
$
\end{Corollary}
\pause

\begin{Proof}
$\bp_{A} = \frac{1}{12}
  \left\{ \big(1 - \sigma(12)\big)\big(1 - \sigma(34)\big)\right\}
 \cdot \left\{\big(1 + \sigma(13)\big)\big(1 + \sigma(24)\big)\right\}$

\pause
\begin{itemize}
\item
  $ \Range(\bp_{A}) \cong F^{[2, 2]}_n$  \quad ($\shape(A) = [2, 2]$)
\pause
\item
  $\Range(\bp_{A}) \subset S^2(\bigwedge^2 \bC^n)$
\end{itemize}
\end{Proof}

\end{frame}

   \section{Orthogonal Decomposition of Curvature Tensors}

%%%%%%%%%%%%%%%%%%%%%% frame 6 %%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Orthogonal Decomposition of Curvature Tensors}

Let $R \in \Curv(\bC^n)$. \quad  

\alert{Ricci curvature:}
%
\vspace{.5ex}

\quad $  \Ric_Q(R)(v, w) =    \sum_{i=1}^n R(e_i, v, e_i,  w)$

\vspace{.5ex}
%
\pause
  \begin{itemize}
  \item 
 Ricci contraction operator $ \Ric_Q: S^2(\bigwedge^2\bC^n) \to S^2(\bC^n) $
\pause
  \item 
      $\Ric_Q$ intertwines ${\rm O}(Q)$ actions on $\Curv(\bC^n)$ and $S^2(\bC^n)$
\pause
  \item
   $\Ric_Q$ is the only nonzero contraction operator on $S^2(\bigwedge^2\bC^n)$ 
  \end{itemize}
\pause

\alert{Scalar curvature:}

\vspace{.5ex}
%
\quad $  s_Q(R) = \tr_Q(\Ric_Q(R)) = \sum_{i,\, j=1}^n R(e_i, e_j, e_i, e_j)$
%
\vspace{.5ex}

\pause
$R \mapsto s_Q(R)$ gives ${\rm O}(Q)$ intertwining operator 
$s_Q: \Curv(\bC^n) \to \bC$ (trivial ${\rm O}(Q)$ module)

\end{frame}

%%%%%%%%%%%%%%%%%% frame 7 %%%%%%%%%%%%%%%%%%%
\begin{frame}

Construct ${\rm O}(Q)$ intertwining operator  $S^2(\bC^n) \to S^2(\bigwedge^2\bC^n)$:

\pause
%
\begin{itemize}
\item
 $S^2(\bC^n) \cong $ $Q$\,-symmetric linear maps of $\bC^n$
%
\ ($Q \leftrightarrow I$)
%
\item
$S^2(\bigwedge^2\bC^n) \cong $ $Q\otimes Q$
  \,-symmetric linear maps of $\bigwedge^2 \bC^n$
\end{itemize}
%
\pause
Given $A, B \in S^2(\bC^n)$, define linear map $A\wedgecirc B$ on $\bigwedge^2 \bC^n$:

%
\vspace{.5ex}
\quad $   (A\wedgecirc B)(v\wedge w) = Av\wedge Bw + Bv \wedge Aw $
\quad\mbox{for $v, w \in \bC^n$}
\vspace{.5ex}
\pause

\begin{itemize}

\item
  $A\wedgecirc B \in S^2(\bigwedge^2 \bC^n)$
\pause

\item
  $A\wedgecirc B$ satisfies Bianchi identity, so $A\wedgecirc B \in \Curv(\bC^n)$
\pause

\item
 $ \Ric_Q(A\wedgecirc Q) = \tr_Q(A)Q + (n-2)A$ \quad \alert{$(\star\star)$}
\pause

\end{itemize}

\alert{$n = 2$:}

\begin{itemize}

\item
  $\dim \Curv(\bC^2) = 1$
\pause

 \item
 $ R = \frac{1}{4}s_Q(R)\, Q\wedgecirc Q$ 
\pause

 \item
 $\Ric_Q(R) = \frac{1}{2}s_Q(R)Q$
\end{itemize}
%
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%  frame 8 %%%%%%%%%%%%%%%%%
\begin{frame}

Assume \alert{$n \geq 3$}.   Given $R \in \Curv(\bC^n)$, set 

\quad $ A = \frac{1}{n-2}\Big\{\Ric_Q(R) - \frac{1}{n}s_Q(R)Q \Big\} \in S^2(\bC^n)$

\vspace{.5ex}

\quad $  C = R - A\wedgecirc Q - \gamma s_Q(R) Q\wedgecirc Q \in \Curv(\bC^n)$ 
 \quad ($\gamma = \frac{1}{n(2n - 1)}$)

\pause
\vspace{.5ex}

The normalizing constants are chosen so that

\begin{itemize}
\item
    $\tr_Q(A) = 0$
\pause
\item
   $\Ric_Q(C) = 0$  \quad by \alert{$(\star\star)$}
\end{itemize}
\pause
%
  \begin{align*} 
   R =&\ \gamma s_Q(R) Q\wedgecirc Q +  A\wedgecirc Q +  C
\\    
   =& \ 
  \mbox{\alert{scalar} part + \alert{traceless Ricci} part + \alert{Weyl} part
   \quad \alert{$(\star\star\star)$} }
  \end{align*}

\pause

Representation-theoretic description of decomposition 
\alert{$(\star\star\star)$}

\vspace{.5ex}

Define  \alert{Weyl conformal curvature tensors}:

\qquad    $\Weyl_Q(\bC^n) = \{ C \in \Curv(\bC^n) \,:\,  \Ric_Q(C) = 0 \} $

\pause

\alert{$Q$-harmonic} (traceless)  symmetric two-tensors:

\vspace{.5ex}
 
\qquad $\cH^2_{\rm sym}(\bC^n, Q) = \{ A \in S^2(\bC^n) \,:\, \tr_Q(A) = 0 \}$

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%  frame 9 %%%%%%%%%%%%%%%%%
\begin{frame}

\alert{Properties} of  $\cH^2_{\rm sym}(\bC^n, Q)$:

\begin{itemize}
\item
 Irreducible under $\SO(Q)$ (\alert{Cartan component} of $\bC^n \otimes \bC^n$)

\pause
\item
  Highest weight $2\varpi_1$ (\alert{$n \neq 4$}) or $2(\varpi_1 + \varpi_2)$
 (\alert{$n = 4$})  
\pause

\item 
   Dimension  = $\frac{1}{2}n(n+1) - 1 = \frac{1}{2}(n+2)(n-1)$ 
\pause

\item 
$\cH^2_{\rm sym}(\bC^n, Q) \hookrightarrow  \Curv(C^n)$ by 
 $A \mapsto A \wedgecirc Q$
\end{itemize}
\pause

\begin{theorem}
The space of curvature tensors decomposes under ${\rm O}(Q)$ as 

\vspace{1ex}
  $\quad \Curv(\bC^n) = \bC (Q\wedgecirc Q) \alert{ \oplus }
    \big(\cH^2_{\rm sym}(\bC^n, Q) \wedgecirc Q\big)
      \alert{\oplus} \Weyl_Q(\bC^n)$
\vspace{1ex}

Hence $ \alert{\dim} \Weyl_Q(\bC^n) = \frac{1}{12}n(n+1)(n+2)(n-3)$.

In particular, Ricci curvature determines curvature when \alert{$n = 3$}.
\end{theorem}

\pause

\begin{proof}

Formula $\alert{(\star\star)} \Longrightarrow $ 
the sum $\alert{(\star\star\star)}$ is direct 
$\Longrightarrow$ dimension formula

\end{proof}

\end{frame}

%%%%%%%%%%%%%%%%%%% frame 10 %%%%%%%%%%%%%%%%

\section{The Space of Weyl Curvature Tensors}

\begin{frame}
     \frametitle{The Space of Weyl Curvature Tensors}

 \alert{Goal:} 

Show $\Weyl_Q(\bC^n)$ is irreducible under ${\rm O}(Q)$ (\alert{$n \geq 4$})

\alert{Method:} 

Find a highest weight vector and use Weyl dimension formula.

\pause
\vspace{.5ex}

Take  \alert{$Q$-isotropic basis} for $\bC^n$: 

\quad $  f_k = \frac{1}{\sqrt{2}}\big(e_k + \sqrt{-1}e_{n+1-k}\big),
\quad
  f_{-k} = \frac{1}{\sqrt{2}}\big(e_k - \sqrt{-1}e_{n+1-k}\big) $

for $k = 1, \ldots, l$ ($l = \left\lfloor \frac{n}{2} \right\rfloor$)
\quad (For $n$ \alert{odd}:   $f_{0} = e_{l+1}$)

Let $F_k = \Span \{f_1, \ldots, f_k \}$ for $k = 1, \ldots, l$ 
(\alert{$Q$-isotropic subspace})
\pause

Take Borel subgroup $B \subset \SO(Q)$ as stabilizer of \alert{isotropic flag}

\qquad $F_1 \subset F_2 \subset \cdots \subset F_l$
  
(include $F_{l+1} = \Span \{f_0, f_1, \ldots, f_l \}$ when $n$ odd) 
\pause

$\varpi_1, \ldots, \varpi_l$ \alert{fundamental} 
 highest weights for $\fso(Q)$

\end{frame}

%%%%%%%%%%%%%%%%%%%%%  frame 11 %%%%%%%%%%%

\begin{frame}

Set $ C_0 = (f_1\wedge f_2)\otimes (f_1\wedge f_2) \in S^2(\bigwedge^2 \bC^n)$

\begin{itemize}
\item
 $ C_0 \in \Weyl_Q(\bC^n)$ 
\pause

\item
  $C_0$ is a $B$ eigenvector of weight $2\varpi_2$ (if \alert{$n > 4$})

   or $4\varpi_1$ (if \alert{$n = 4$}) 
\pause

\item
  $V_n = \Span \SO(Q)\cdot C_0$ is  irreducible under $\SO(Q)$   
 (theorem of the highest weight)
\pause
\end{itemize}

\begin{theorem}
Assume \alert{$n > 4$}.  $\Weyl_Q(\bC^n)$ is irreducible under $\SO(Q)$
and has highest weight $2\varpi_2$.
\end{theorem}

\pause

\begin{proof}
Have $ V_n \subset \Weyl_Q(\bC^n)$, and Weyl dimension formula $\Longrightarrow$ 

\vspace{.5ex}

\quad $\dim V_n = \frac{1}{12}n(n+1)(n+2)(n-3) = \dim \Weyl_Q(\bC^n)$
\end{proof}

% note not correct for n = 6
% \alert{Note:} $\varpi_2$ is the highest weight of $\bigwedge^2 \bC^n$ if $n > 4$.

\end{frame}

%%%%%%%%%%%%%%%%  frame 12 %%%%%%%%%%%

\begin{frame}

Assume \alert{$n = 4$}:

\alert{Special feature:} $\fso_{4} \cong \fsl_2 \oplus \fsl_2$ (semisimple)

\pause
Let $\tau \in {\rm O}(\bC^4, Q)$ fix $f_{\pm1}$
and interchange $f_{2} \leftrightarrow f_{-2}$ 
\quad \alert{($\det \tau = -1$)}

Set $ \overline{C_0} = \tau \cdot w = (f_1\wedge f_{-2}) \otimes (f_1\wedge f_{-2})
 \in \Weyl_Q(\bC^4)$

\pause

\begin{itemize}

\item
 $\overline{C_0}$  eigenvector for $B$ of weight $4\varpi_2$
\pause

\item
 $\overline{V}_4 = \tau \cdot V_4 \subset \Weyl_Q(\bC^4)$ is
irreducible under $\SO(Q)$
\pause


\end{itemize}

\begin{theorem}

$\Weyl_Q(\bC^4) = V_4 \alert{\oplus} \overline{V}_4$
and is irreducible under ${\rm O}(\bC^4, Q)$.

\end{theorem}

\pause
\begin{proof}
  $ \tau:  V_4 \leftrightarrow \overline{V}_4  $ and $\dim V_4 = \dim \overline{V}_4 = 5$
while $\dim \Weyl_Q(\bC^4) = 10$
\pause

  $V_4 \cap \overline{V}_4 = 0$ (\alert{inequivalent} for $\SO(Q)$)
\end{proof}
\pause

\alert{Note:} Likewise, $\bigwedge^2 \bC^4$ is irreducible under ${\rm O}(Q)$, but
decomposes under $\SO(Q)$ with highest weights $2\varpi_1$ and
$2\varpi_2$.

\end{frame}

%%%%%%%%%%%%%%%%%% frame 13 %%%%%%%%%
     \section{Conformal Change of Metric  Tensor}

\begin{frame}
     \frametitle{Conformal Change of Metric  Tensor}

Replace $g$ by $\alert{\widetilde{g}} = e^{2f}g$ \quad where $f \in \cC(M)$. 

Orthogonal group ${\rm O}(g) = {\rm O}(\widetilde{g})$ is unchanged.

\vspace{.5ex}

\pause

\alert{Problem:} Determine the change in the Weyl, traceless Ricci, and scalar
parts of the Riemann curvature tensor.

\vspace{1ex}

\pause
\alert{New Riemannian connection:} 

\vspace{.5ex}

\quad  $\alert{\widetilde{\nabla}}_X Y = \nabla_X Y + \Phi(X,Y)$
%
\quad with $\Phi(X,Y) = \Phi(Y,X) \in \cT(M)$ 

\pause

\vspace{.5ex}

Explicit formula:

\vspace{.5ex}

 \quad  $\Phi(X,Y) = df(X)Y + df(Y)X - g(X,Y)Df$  \quad ($Df = \grad_g f$)

\vspace{.5ex}

\pause

[Follows from 
%
 $e^{-2f}X(e^{2f}g(Y,Z)) = 
 g(\alert{\widetilde{\nabla}}_X Y, Z) + g(Y,\alert{\widetilde{\nabla}}_XZ)$
%
and cyclic permutation of $X, Y, Z$.]

\end{frame}

%%%%%%%%%%%%%%%%%% frame 14 %%%%%%%%%

\begin{frame}

\alert{New Curvature Tensor:}

\vspace{.5ex}

\quad $  \alert{\widetilde{R}} = e^{2f}R -  e^{2f}Q \wedgecirc A$ 
%
\quad with $A \in S^2(\bC^n) \cong S^2(\bC^n)^{*}$ via $Q$
 
% check sign

\pause
Explicit formula: 


\quad   $ A = D^2f - df\otimes df + \frac{1}{2} |Df|^2 Q$
%
\quad (\alert{long} calculation)
\vspace{.5ex}

 where\quad $D^2f(X,Y) = XY(f) - (\nabla_X Y)(f)$ 
  \quad (\alert{$Q$-Hessian} of $f$)

\quad\qquad $|Df|^2 = Q(Df, Df)$
\pause

\alert{Conclusion:} Under conformal change of metric $g \to e^{2f}g$

\begin{itemize}
%
\item
 Weyl part of the curvature is multiplied by $e^{2f}$.
%
\pause
\item
 Traceless Ricci curvature is modified by adding term
%
\vspace{.75ex}
%
\quad $  (n-2)D^2f - (n-2)(df\otimes df) 
 + \frac{n-2}{n}\big\{ \Delta f + |Df|^2 \big\} Q $

\vspace{.75ex}
%
where \quad $\Delta f = -\tr_Q(D^2 f)$ \quad is the \alert{$Q$-Laplacian}.

\pause
\item
 Scalar curvature is multiplied by $e^{2f}$ plus  term
\vspace{.5ex}

\quad $e^{2f}\big\{ 2(n-1)\Delta f - (n-1)(n-2)|Df|^2 \big\}$
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%% frame 15 %%%%%%%%%

\section{Further Topics}

\begin{frame}

\frametitle{Further Topics}

\begin{itemize}
\item
Spaces of Covariant Derivatives $\nabla R, \nabla^2 R, \ldots$  

Decomposition as representation spaces for orthogonal group
(Kulkarni, Strichartz)  

\pause

\item
Structure of Weyl Tensors


\begin{itemize}
\item
$n = 4$: Petrov (1954)  $n > 4$: Coley et al. (2004) 

Important for study of \alert{gravity waves}

\pause

\item 
\alert{Problem:} Describe classification in terms of structure of orbits
of $G = \SO(n, Q)$ on $V = \Weyl_Q(\bC^n)$.

\pause

\item
Highest weight vector orbit:
$X = \overline{G\cdot C_0} = G\cdot C_0 \cup \{0\} \subset V$  

\pause
$X$ is a multiplicity-free $G$ space (Vinberg--Popov)

$X/\bC^{\times} = G/P$  with
%
$P = $ maximal parabolic $\longleftrightarrow \varpi_2$

\pause

\item
Find other $G$ orbits on $V$ and orbit invariants (Strichartz)

$n=4$: Use classical invariant theory of binary quartics

   \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}

\alert{Hermann Weyl:} 

\quad

``The wish to understand what really is the
mathematical substance behind the formal apparatus of relativity theory led me
to the study of representations and invariants of groups.''

\quad

{\em Relativity theory as a stimulus in mathematical research} (1949)

\end{frame}


\end{document}


