Let $S$ be a (n embedded) submanifold of a manifold $M$. Let $\Phi(M)$ be a diffemorphism. Is $\Phi(S)$ an embedded submanifold?

Question

I feel like the answer to the above question is "yes", as one should just be able to compose the embedding with the diffeomorphism to get an embedding, but I would appreciate confirmation, as I still find some of this stuff a bit tricksy

Answer 1

If $\Phi:M\to N$, is a diffeomorphism of smooth manifolds, and $i:S\to M$ is inclusion, then consider $F=\Phi\circ i:S\to N$. Clearly, $F$ is a diffeomorphism onto its image.

Fix a point $y \in F(S)$. Then, $y = F(x)$ for some $x\in S.$ There is a slice chart $(U,\phi)$ about $x$ such that $U=V\cap S$ for some open set $V$ in $M.$ Then, since $F(V)$ is open in $N,\ (\phi\circ F^{-1},F(U))$ is a slice chart about $F(y)$, where $F(U)=F(V)\cap F(S).$

This shows that $F(S)$ is an embedded submanifold of $N$.