Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 25I / c / Solution

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.

Solved by gpt-5.6-sol high.