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Suppose $N$ is an immersed submanifold of a smooth manifold $M$, such that $N$ has the subspace topology, and is a closed set in $M$. Let $n = \mathrm{dim}\ N$ and $m = \mathrm{dim}\ M$.

Given any $x \in N$, the problem is to show that there exists a chart $( U, \varphi)$ for $M$ around $x$, such that $( U \cap N, \varphi(x_1, \ldots, x_n))$ is a chart for $N$ around $x$.

The clue is that this is supposed to follow from the implicit function theorem. For ease in typing, let me refer to the statement in wikipedia, reproduced in this question.

I am wondering how to prove the aforementioned statement, from the existence of the function $g$.

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Rolanda Hooch
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  • Should this read, "Suppose $N$ is an embedded submanifold ..." ? Or do you really mean immersed? – Matt Mar 02 '13 at 18:40
  • @Matt, Rolanda: I think, with those hypotheses, $N$ doesn't need to be closed. Look at proof of Theorem 5.8 in Lee's "Introduction to Smooth Manifolds", where it's shown that the subspace has slice charts. It is enough to know that the subset has the subspace topology and the inclusion is an immersion, for the existence of slice charts. Am I right? – Weltschmerz Nov 12 '14 at 02:40

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First of all we have to specify what a submanifold is. In fact by what you have written above the smooth structure of $ N $ is not specified (then $ N $ is merely a topological subspace).

If $ N $ is a smooth manifold with $ dim N=n$ an embedded m-submanifold is the image $ f(M) $ of an embedding $ f:M \rightarrow N $ where $ M $ is a smooth manifold of dimension $ m \leq n $. This is merely a definition and $ f(M) $ is merely a subset of $ N $. Now it holds (see theorem 8.3 of 'Introduction to smooth manifolds', Lee):

Theorem For every $ p \in f(M) $ there exists a neighbourhood $ U \subset N $ of $ p $ and a smooth cooordinate chart $ \varphi:U \rightarrow R $ such that $ \varphi(U \cap f(M)) $ is a slice of $ \varphi(U) $ (see Lee's book for the definition of slice, that perhaps you already know).

Using this theorem you can prove that (see Theorem 8.2 of Lee)

Theorem An embedded submanifold with the subspace topology is a topological manifold of dimension $ m $. Moreover there exists a unique smooth structure on $ f(M) $ (obtained by charts of the previous theorem; see proof for understanding this fact) such that the inclusion $ i:f(M)\rightarrow N $ is an embedding.

If $ f:M \rightarrow N $ is merely an immersion (injective or not) the first theorem cited above is not always true (think about the 'eight figure' as immersed subamnifold in $ R^2 $). Then perhaps it is not always true that the image of an immersion is a topological manifold if we consider it as a topological subspace of the ambient manifold.