Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 2 / f / Solution

Let $I(N)$ be the vanishing ideal of an embedded submanifold. Tangency says $X(I(N)),Y(I(N))\subseteq I(N)$. Hence, for $h\in I(N)$,so the criterion from part (d) makes $[X,Y]$ tangent to $N$.

In adapted coordinates, the tangential coefficients of the displayed bracket use only the restrictions of the tangential coefficients of $X,Y$ and their derivatives along $N$; all normal coefficients vanish there. Consequentlyso the restriction depends only on $X{|}_N$ and $Y{|}_N$. This is tangency under the Lie bracket.