Given $E:N\to M$ an embedding and $V,W\in \mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.

I have encounter some difficulties while looking at an exercise online. It basically goes as follows:

Given $$E:N\to M$$ an embedding and $$V,W\in \mathfrak{X}(M)$$ tangent to $$N$$, we claim that the commutator of $$V$$ and $$W$$ is also tangent to $$N$$.

I would like to have some ideas about how to attack the problem effectively.

• There are a few possible approaches, depending on your definition of the commutator. – Amitai Yuval Jan 23 at 6:49
• It is just the usual one: $[A,B]=AB-BA$. – DaveWasHere Jan 23 at 11:35

If $$V$$ and $$W$$ are tangent to N, it means that there are vector fields $$v$$ and $$w$$ in $$\mathfrak X(N)$$ such that for any $$x\in N$$ we have $$V_{E(x)}=E_*v_x$$ and the same is true for $$W$$. To be able to interpret things properly, assume that $$V$$ and $$W$$ are smoothly extended off $$E(N)$$.
Then $$v$$ and $$V$$ are $$E$$-related and so are $$w$$ and $$W$$.
But we know that for $$E$$-related vector fields the commutators are also $$E$$-related, so we have (restricted to $$E(N)$$) $$[V,W]=E_*[v,w],$$
• Thanks for the comment! But where do we use the fact that $\mathfrak{X}(M)\ni V,W$? – DaveWasHere Jan 23 at 15:51