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I was doing Exercise 8.3 from the book From Calculus to Cohomology, Ib H. Madsen,

Suppose that $M \subset \mathbb{R}^k$ (with the induced topology from $\mathbb{R}^k$) is an $n$-dimensional topological manifolds. Include $M$ in $\mathbb{R}^{k+n}$. Show that $M$ is locally flat in $\mathbb{R}^{k+n}$.

What is the purpose of including $M$ in $\mathbb{R}^{k+n}$, when M is already an embedded submanifold of $\mathbb{R}^k$?

Is it generally true that a smooth submanifold is always a locally flat submanifold? I find that the following definition of smooth submanifold in the book seems to coincide with the definition of local flatness on Wikipedia.

A subset $N \subset M^n$ of a smooth manifold is said to be a smooth submanifold if the following condition is satisfied: for every $x \in N$ there exists a chart $h: U\rightarrow U'$ on $M$ such that $x \in U$ and $h(U \cap N)=U'\cap \mathbb{R}^k$.

Jeffrey Mak
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  • What is the definition of locally flat for topological manifolds? And should it be true for all $n>0$? – Paul Frost Feb 25 '24 at 19:09
  • @PaulFrost Local Flatness had not been introduced in the book. And no, it specified that $n$ be the dimension of $M$. The only idea I have is that $M$ is locally a graph in $\mathbb{R}^{n+k}$ if inclusion is done correctly. – Jeffrey Mak Feb 25 '24 at 20:12
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    For local flatness of topological submanifolds in the case $n=1, k=3$ see my answer here. I think similar arguments work in general. And smooth submanifolds are always locally flat. It is a good exercise in understanding the definition of a smooth submanifold and the constant rank theorem (depending on your definition of a smooth submanifold). – Moishe Kohan Feb 26 '24 at 04:31
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    The concept of a locally flat topological submanifold is a bit hidden in the text. It is introduced before Theorem 8.14: "It is defined in analogy to Definition 8.8, but with a homeomorphism instead of the diffeomorphism $h$. "

    This is exactly the Wikipedia definition of local flatness. Therefore a smooth submanifold of a smooth manifold is a locally flat submanifold by definition.

     – Paul Frost Feb 26 '24 at 10:00
  • @PaulFrost To sum it up, not all topological submanifolds are locally flat, but every smooth submanifold is locally flat, due to the constant rank theorem. Am I correct? – Jeffrey Mak Feb 26 '24 at 17:40
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    @JeffreyMak Yes, each smooth submanifold is locally flat. But this has nothing to do with the constant rank theorem, it is the definition of a smooth submanifold. Okay, there are alternative definitions of smooth submanifolds (see here), and local flatness would be a theorem then. The Alexander horned sphere is an example of a topological manifold $M \subset \mathbb R^3$ which is not locally flat. – Paul Frost Feb 26 '24 at 18:11

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