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I believe that more or less this type of question has already appeared but when typing this question, I cannot find it. Nevertheless, this question concerns two types of definition of (smooth) manifolds: one for submanifolds of $\mathbf R^n$ and the other for abtract manifolds. Obviously, these definitions are generally not equivalent but I want to understand these in the situation of embedded smooth submanifolds of $\mathbf R^n$.

Let me recall these definitions for clarity.

Definition 1 (Munkres, etc). A subspace $M \subset \mathbf R^n$ is a $k$-manifold of class $C^r$ if for any point $p \in M$, there is an open set $V \subset M$, an open set $U \subset \mathbf R^k$ and a continuous map $\alpha : U \to V$ such as

  1. $\alpha \in C^r$ (in the extended sense)
  2. $\alpha^{-1}: V \to U$ is continuous
  3. $D\alpha$ has rank $k$ at any point in $U$.

Note that the continuity of $\alpha^{-1}$ in 2. is enough because $\alpha^{-1}$ is in fact of class $C^r$; see Thm 24.1 in Munkres's book. So, basically Definition 1 says that for any point $p \in M$, there is a diffeomorphism between an open set containing $p$ and an open subset in $\mathbf R^k$.

Definition 2 (Lee, etc). A smooth manifold $M$ of dimension $k$ is a topological space equipped with a smooth structure, which basically says that there is a family of local charts $(U, \phi)$ of $M$, where $U$ is open in $M$ and $\phi$ is a homeomorphism from $U$ onto an open subset of $\mathbf R^k$ such that the translation maps between two overlapped charts are smooth. (I do not want to write down in details because more or less this is standard and appears in many textbooks.)

Let us now consider an embedded smooth submanifold $M \subset \mathbf R^n$ of dimension $k$. (By definition, this means that the inclusion $i : M \to \mathbf R^n$ is a smooth embedding but here $M$ is a subset of $\mathbf R^n$ for simplicity.) My question is how to verify the three conditions in Definition 1. I hardly see how to construct $\alpha$ with enough regulariry around a given point since $\phi$ and $\phi^{-1}$ in Definition 2 are only continuous.

QA Ngô
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  • The confusion here is that $\phi$ is only required to be continuous because there is not smooth structure on $M$ before there are charts. But, what we know is that because the inclusion is differentiable and so we require that $i\circ \phi = \phi$ is differentiable for every $\phi$. Essentially, requiring that $i$ is in fact an embedding is what is doing the work here. – memerson Mar 20 '21 at 08:14
  • Are you saying that a priori $\phi$ is only required to be continuous, however, to be able to embed into $\mathbf R^n$, the map $\phi^{-1}$ is actually smooth? (From this $\phi$ is also smooth in the extended sense. Hence, any coordinate map from the smooth structure of an embedded submanifold of $\mathbf R^n$ is in fact smooth in the extended sense.) – QA Ngô Mar 20 '21 at 16:46

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