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 $N$.
In fact, diffeomorphic manifolds are "the same", but drawn differently. A diffeomorphism is a dictionnary between them. Thus, every geometric property of one is translated into a geometric property of the other, and bijectively.