Solution
= Solution
Part (b) already shows $\exp(U\cap\mathcal A)\subset V\cap O(n)$. Conversely, if $Q\in V\cap O(n)$ and $A=L(Q)\in U$, then
$$
\exp(A^T)=Q^T=Q^{-1}=\exp(-A).
$$
Both $A^T$ and $-A$ lie in $U$, where the <matrix exponential> is injective. Therefore $A^T=-A$, proving the reverse inclusion. The restricted map is a <bijection>, indeed a <diffeomorphism> between these local sets.