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.

Solution

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

 Solution (source code)