Math 6510 • Differentiable Manifolds

Consider the Lie Group
\(G=\Ogrp(n)=\{A\in\mathcal{M}_n:A^TA=I\}\) where
\(\mathcal{M}_n\cong\R^{n^2}\) is the collection of real \(n\times n\) matrices and
\(A^T\) is the transpose of \(A\).
\begin{enumerate}
\item[(a)] Describe the set of matrices which form the tangent space \(T_PG\) at the
point \(P\in G\).
\item[(b)] Given \(V\in T_IG\) and \(P\in G\), find \(X_V(P)\in T_PG\) where \(X_V\) is
the left invariant vector field on \(G\) such that \(X_V(I)=V\). Show that your \(X_V\)
is left invariant.
\item[(c)] Let \(\theta_t^X\) be the flow for the left invariant vector field
\(X=X_V\) for some \(V\in T_IG\). Find and verify a formula for \(\theta_t^X(P)\) in
terms of \(P\), \(t\) and \(V\). Compute \(\theta_t^X(I)\) in case \(G=O(2)\) and
\[
  V=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.
\]
\end{enumerate}