Math 6510 • Differentiable Manifolds

Let \(\mathcal{M}(n)\) denote the space of all \(n\times n\) matrices with real entries, iden-
tified with \(\mathbb{R}^{n^2}\), and let \(S(n)\) be the subspace of symmetric matrices.
Consider the smooth map \(F:\mathcal{M}(n)\to S(n)\) defined by \(f(A)=AA^{\top}\),
where \(A^{\top}\) is the transpose of \(A\).
\begin{enumerate}

\item
Show that the identity matrix \(I\) is a regular value of \(f\) .


\item
Deduce that the orthogonal group

\[
O(n)=\{A\in\mathcal{M}(n)\mid AA^{\top}=I\}
\]


is a submanifold of \(\mathcal{M}(n)\). Carefully state any theorems you are
citing.

\end{enumerate}