# Tangent vector field to embedded submanifold $S \subset M$ induces unique $\iota$-related vector field on $S$.

Let $$S\subset M$$ be an embedded submanifold. A vector field $$X \in \mathfrak{X}(M)$$ is said to be tangent to $$S$$ if $$X_p \in T_pS$$ for all $$p \in S$$. Show that for $$Y \in \mathfrak{X}(M)$$ tangent to $$S$$, there is a unique smooth vector field $$X \in \mathfrak{X}(S)$$ such that $$X$$ is $$\iota$$-related to $$Y$$.

I have seen that since each $$Y_p \in \iota_{*p}(T_pS) \equiv T_pS$$, we can assign tangent vectors $$X_p \in T_pS$$ pointwise, with $$Y_p = \iota_{*p}X_p$$. I'm having trouble showing that the resultant vector field $$X$$ is indeed smooth and unique.

One criterion is checking whether $$Xf \in C^\infty(S)$$ for all $$f \in C^\infty(S)$$. I am aware of the characterization $$T_pS = \{X \in T_pM: Xf = 0 \;\text{ whenever } f \in C^\infty(M)\text{ and } f\big|_S = 0\},$$ but I don't quite see if this helps.

Let $$f\in C^\infty (S)$$. We need only to show that $$Xf \in C^\infty(S)$$. Since smoothness is a local property, it suffices to check it locally around each $$p\in S$$.

Let $$p\in S$$. Let $$U$$ be an open neighborhood of $$p$$ in $$M$$, so that $$f$$ extends to a smooth function $$F$$ on $$U$$ (indeed one can choose $$U$$ so that $$M\subset U$$, see here). Since $$Y$$ is smooth, $$YF$$ is smooth. But since

$$YF|_{S\cap U} = Xf|_{S\cap U}$$

(by definition of the action of vector fields as directional derivative), it implies that $$Xf|_{U\cap S}$$ is smooth. since this holds for all $$p\in S$$, $$Xf$$ is smooth for all smooth $$f$$ on $$S$$. Hence $$X$$ is a smooth vector field.