Vector fields on submanifolds

Question

I struggle with how to apply a vector field on a submanifold on some smooth function on the whole manifold, i.e.: Let $(M,g)$ be a riemannian manifold, $N \subset M$ a submanifold. Now let $X \in \mathfrak{X}(N)$ and $f \in C^{\infty}(M)$.

What exactly is $X(f)$ then? If $f$ is zero on $N$, is $X(f)=0$ then?

Answer 1

Within an $\varepsilon$ of a doubt, I'm pretty sure that $X(f)$ is shorthand for $X(i^*f)$, where $i:N\to M$ is the inclusion.

In this case, if $f$ is zero on $N$, then $i^*f=0\in C^\infty(N)$ and thus $X(f)=0$.

Answer 2

One way to do this is, assuming that $N\subseteq M$ is an embedded (or immersed, I guess would also work) submanifold, is to note that the inclusion $i$ is a smooth immersion so $di_p:T_pN\to T_pM$ is injective and we may thus identify $T_pN$ as a subspace of $T_pM$.

So, if $v\in T_pN$, then $di_pv\in T_pM$ is defined by $di_pv(f)=v(f\circ i)$, which just says that $di_pv$ acts on smooth functions from $M$ by letting $v$ act on the restriction of $f$ to $N$, which is smooth because $f$ is and $i$ is.