# Submanifolds are preserved under diffeomorphisms

I've been reading books about differential manifolds. There is a remark in the book that says

Remark: Submanifolds are preserved under diffeomorphism, i.e If $$A$$ is a submanifold of $$M$$ and $$\tau : M \rightarrow N$$ is a diffeomorphism, then $$\tau(A)$$ is a submanifold of $$N$$.

Can anyone enlighten me why is this true or is there post before regarding to this fact.
Any help would be much appreciated!

Recall that being a submanifold is a local property: if $$M^n$$ is a manifold and $$Y \subset M$$, then $$Y$$ is a submanifold of $$M^n$$ of dimension $$m$$ if for every point $$p\in Y$$, there exists an open subset $$U$$ in $$M$$ with $$p\in U$$ and a chart $$\varphi: U \to \mathbb{R}^n$$ such that $$\varphi\left(Y\cap U \right) = V^m\cap \varphi(U)$$ where $$V^m$$ is a linear subspace of $$\mathbb{R}^n$$, of dimension $$m$$. This says that there exists local charts such that in those charts, $$Y$$ is an a linear subspace of the chart.
Now, if $$f : M \to N$$ is a diffeomorphism, and if $$Y \subset M$$ is a submanifold, take a chart $$\varphi : U \subset M \to \mathbb{R}^n$$ that is used in the definition for $$Y$$ to be a submanifold. Let $$\tilde{Y} = f(Y) \subset N$$,$$\tilde{U} = f(U) \subset N$$ and define $$\tilde{\varphi} : \tilde{U} \to \mathbb{R}^n$$ to be $$\tilde{\varphi}(p) = \varphi(f^{-1}(p))$$. It is clear that $$\tilde{\varphi}$$ is a chart in which we can read that $$\tilde{Y}$$ is a submanifold! We have then shown that $$\tilde{Y}$$ is a submanifold of $$N$$. So every submanifold of $$M$$ gives birth to a submanifold of $$N$$. Conversly, as $$f^{-1}: N \to M$$ is also a diffeomorphism, every submanifold of $$N$$ gives birth to a submanifold of $$M$$, and there is a correspondance between the set of submanifolds of $$M$$ and the set of submanifolds of $$N$$.
• Question is $\tilde{Y}$ and $\tilde{U}$ the same ? Commented Oct 28, 2020 at 13:27
• No. $\tilde{Y}$ is the image of $Y$ under the diffeomorphism $f$, while $\tilde{U}$ is the image of the open subset $U$ under the same diffeomorphism. I've edited the typo! Sorry Commented Oct 28, 2020 at 13:33