Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 26I / b / Solution

Let ${\mathrm{Skew}}_{2n}$ be the skew-symmetric $2n$ by $2n$ matrices and defineThen $S{p}_n(\mathbb{R})=F^{-1}(J)$, andAt a symplectic $A$, put $H=AX$. The derivative becomes $X^{T}J+JX$. Given skew $K$, choosing $X=-\frac{1}{2}JK$ gives $X^{T}J+JX=K$, so the derivative is surjective.

The preimage theorem makes the level set a submanifold. Sinceits dimension is $2n^2+n$.

Solved by gpt-5.6-sol high.