Past exam of the short options course of the University of Oxford / 2023 / 4 (Multidimensional analysis and geometry)

For the operator norm, submultiplicativity bounds the remainder of the matrix exponential byTherefore $\Vert \exp A-\exp 0-A\Vert /\Vert A\Vert \to 0$. This is precisely Fréchet differentiability at zero, with

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

The matrix exponential is smooth, and part (a) identifies its derivative at zero as the identity. The inverse function theorem therefore gives a local diffeomorphism from an open neighborhood $U_0$ of zero onto a neighborhood of $I$. Choose a sufficiently small operator norm ball $U\subset U_0$ and let $V=\exp U$. This is again a diffeomorphism onto an open set, since it is a restriction of the original one.

The ball $U$ is preserved by $A\mapsto A^T$ and $A\mapsto -A$. Since $(\exp A{)}^T=\exp (A^T)$ and $(\exp A{)}^{-1}=\exp (-A)$, $V$ is preserved by adjoint and inversion. For its local inverse $L$, injectivity also gives

Part (b) already shows $\exp (U\cap \mathcal{A})\subset V\cap O(n)$. Conversely, if $Q\in V\cap O(n)$ and $A=L(Q)\in U$, thenBoth $A^T$ 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.

The linear map $\pi (A)=(A+A^T)/2$ is the projection onto the space $\mathcal{S}$ of symmetric matrices, with kernel $\mathcal{A}$. Define $F=\pi \circ L$ on $V$. Since $DL$ is invertible everywhere on $V$ and $\pi$ is onto, $DF$ is onto everywhere. By part (ii),The regular level set theorem makes this a submanifold of codimension $\dim \mathcal{S}=n(n+1)/2$, and henceThis neighborhood is $V$ around $I$; the source's final occurrence of $U$ in this step is a neighborhood-label slip. For any $Q\in O(n)$, left multiplication by $Q$ is an invertible linear map of the ambient matrix space and carries $V\cap O(n)$ onto $QV\cap O(n)$. These translated charts establish the submanifold property at every group element.