Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 25J (Differential Geometry)

A subset $X\subset {\mathbb{R}}^N$ is a smooth $k$-manifold if every $x\in X$ has a neighborhood parametrized by a smooth map $\varphi :U\subset {\mathbb{R}}^k\to X$ that is a homeomorphism onto its image and has derivative of rank $k$. DefineIf $\psi$ is another parametrization, the transition maps satisfyThe transition derivative is invertible, so the two images coincide. Thus the tangent space is independent of the parametrization.

The determinant is continuous, sois open in ${\mathbb{R}}^{n^2}$. It is therefore a smooth manifold of dimension $n^2$, and

The differential of determinant at $A$ isIt is nonzero at every $A\in SL(n,\mathbb{R})$, so the regular-level-set theorem makes $SL(n,\mathbb{R})$ a manifold of dimension $n^2-1$. At the identity,

Consider $F(A)=A^{T}A$ with values in the symmetric matrices. At the identity,which is onto the $n(n+1)/2$-dimensional space of symmetric matrices. The same rank holds at every orthogonal matrix. Thus $O(n)=F^{-1}(I)$ is a manifold, and its determinant-one component, the special orthogonal group, has dimensionThe tangent equation at the identity is $X^T+X=0$, so