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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 $V\subseteq \mathbb{R}^n$ 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?

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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.

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