Past exam of the short options course of the University of Oxford / 2023 / 4 / b

The curve-based tangent space at $x$ consists of velocities ${\gamma }^{\prime }(0)$ of smooth curves $\gamma :(-\epsilon ,\epsilon )\to M$ satisfying $\gamma (0)=x$. For an arbitrary subset these velocities need not form a vector space; for a smooth submanifold they do, and this is its usual tangent space. This definition suffices for the direct computation at the identity.

If $Q(t)$ is a smooth curve of orthogonal matrices with $Q(0)=I$, differentiating $Q(t{)}^{T}Q(t)=I$ gives $Q^{\prime }(0{)}^T+Q^{\prime }(0)=0$. Thus each tangent velocity is a skew-symmetric matrix.

Conversely, if $A^T=-A$, the curve $Q(t)=\exp (tA)$ satisfiesusing commutation of $-tA$ and $tA$. By part (a), $Q^{\prime }(0)=A$. Both inclusions prove