Krdytkyu

The definition of a submanifold of $mathbb{R}^n$ I was given is the following:

A subset $M subseteq mathbb{R}^n$ is called an $m$-dimensional submanifold of $mathbb{R}^n$ if for every point $x in M$ there exists an open set $U subseteq mathbb{R}^n$ containing $x$ and an open subset $Vsubseteq mathbb{R}^n$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^m times {0})$ with $0 in mathbb{R}^{n-m}$.

I don't quite see why this captures the notion of "a manifold of dimension $m$ locally looks like $mathbb{R}^m$". If I was asked to make this notion rigorous, I'd define a submanifold of $mathbb{R}^n$ as follows:

A subset $M subseteq mathbb{R}^n$ is called an $m$-dimensional submanifold of $mathbb{R}^n$ if for every point $x in M$ there exists an open set $U subseteq M$ in the subspace topology and an open set $V subseteq mathbb{R}^m$ which is diffeomorphic to $U$, that is there exists a diffeomorphism from $U$ to $V$.

Is there a difference between the two definitions? Isn't $mathbb{R}^m times {0})$ diffeomorphic to $mathbb{R}^m$? Does it have something to do with wanting the Jacobi-matrix of the diffeomorphism to be invertible?

Usually, before introducing the notion of manifolds, the concept of what it means to be a smooth map $f colon U rightarrow V$ (or a diffeomorphism) is defined only when $U,V$ are open subsets of $mathbb{R}^n / mathbb{R}^m$. The problem with your second definition is that you want to say a small piece of $M$ looks like a small piece of $mathbb{R}^m$ "in a smooth way" and so you say "there exists a diffeomorphism between $U subseteq M$ and $V subseteq mathbb{R}^m$. However, because $U$ is an open subset of $M$ (in the subspace topology), it won't generally be an open subset of $mathbb{R}^n$ and so it is not clear a priori what it means to be a diffeomorphism between $U$ and $V$.

This can actually be fixed and leads directly to the first definition. Let's say that two subsets $A,B subseteq mathbb{R}^n$ are diffeomorphic if one can find open neighborhoods $A subseteq U$ and $B subseteq V$ (where $U,V$ are open in $mathbb{R}^n$) and a diffeomorphism $phi colon U rightarrow V$ such that $phi(A) = B$. What this definition means in practice is that $A,B$ are diffeomorphic if $A$ can be mapped to $B$ bijectively by a map $phi$ in such a way that it possible to extend this map $A$ to an open neighborhood of $A$ so that the extension is a diffeomorphism (onto some open neighborhood of $B$).

Having this definition in mind, your second definition makes sense and is actually equivalent to the first definition. Why? Well, in the second definition you have the subset $M cap U$ which is open in $M$ (but generally not open in $mathbb{R}^n$) and the subset $V cap (mathbb{R}^m times { 0 })$ which is open in $mathbb{R}^m times { 0_{n-m} }$ (so under the identification $mathbb{R}^m cong mathbb{R}^m times { 0_{n-m} }$ you can think of it as an open subset of $mathbb{R}^m$) but generally not open in $mathbb{R}^n$. The first definition requires you to find a map $phi colon U rightarrow V$ which is a diffeomorphism between open sets which sends $U cap M$ to $V cap (mathbb{R}^m times { 0 })$ and according to the definition I gave this means that $M cap U$ and $V cap (mathbb{R}^m times { 0 })$ are diffeomorphic.