Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 1 / b / Solution

Let $\Omega$ orient $M\times \mathbb{R}$ and write $t$ for the coordinate on $\mathbb{R}$. Along $M\times \{0\}$, contraction with the transverse vector ${\partial }_t$ givesFor every basis of $T_{p}M$, adjoining ${\partial }_t$ gives a basis of $T_{(p,0)}(M\times \mathbb{R})$, so $\omega$ never vanishes. The top form $\omega$ orients $M$.