Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 25I (Differential Geometry)

For a local parametrization $\varphi :U\subset {\mathbb{R}}^k\to X$, the map $(q,w)\mapsto (\varphi (q),d{\varphi }_{q}w)$ parametrizes $TX$ locally. Its inverse uses the base chart and derivative, and its rank is $2k$. These charts make $TX$ a $2k$-manifold.

A value $y$ is regular when $d{f}_x:T_{x}X\to T_{y}Y$ is surjective for every $x\in f^{-1}(y)$. The preimage theorem says $f^{-1}(y)$ is then a submanifold of dimension $\dim X-\dim Y$.

Define $F:TX\to \mathbb{R}$ by $F(p,v)=I_p(v,v)$. On $F^{-1}(1)$, the vertical derivative in direction $v$ is $2I_p(v,v)=2$, so 1 is regular. Since $\dim TX=4$, the preimage theorem makes the unit tangent bundle $SX$ a 3-manifold.