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 $N$.

Question

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.

Thank you in advance!

Answer 1

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 $E(N)$) $$ [V,W]=E_*[v,w],$$

Implying that the commutator is tangent and is independent of the extensions.