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

If $X$ is tangent to $N$ and $f{|}_N=0$, then the restriction of $f$ to every curve in $N$ is zero, so $(Xf){|}_N=0$.

Conversely, use a slice chart for an embedded submanifold. The functions $x^{n+1},\dots ,x^m$ vanish on $N$. Writing $X=X^i{\partial }_i$, the hypothesis givesThus $X$ has no normal component and is tangent to $N$. Equivalently, $X$ preserves the vanishing ideal of an embedded submanifold.