Pardon if this is an duplicate question. Say, for instance that I an embedded submanifold $N$ of a manifold $M$. I also know that $M$ is isomorphic (diffeomorphic?) to $M'$ via the isomorphism $f$. The embedded submanifold $N$ is defined via the regular level set theorem (see for instance Lee, ISM first edition, Corr. 8.10). Can I say that $N$ also is embedded in $M'$?

My attempt: We know that there is an immersion $F:N \to M$, and that this immersion is a topological embedding. It feels as if we could consider $f \circ F : N \to M' $, and while I see that this is an immersion, I fail to see that it is also a topological embedding. Is this not the way to go? Are there other things I can say?

  • $\begingroup$ The map $f \circ F$ send $N$ homeomorphically onto $f(F(N))$ since $N$ homeomorphic with $F(N)$ through $F$ and $f$, in particular, is a homeomorphism. $\endgroup$ – Kelvin Lois Oct 1 '20 at 15:14
  • $\begingroup$ Thank you for your answer! So, perhaps a stupid question, how does this give that it is a topological embedding? And also, the embedding into $M$ induces a structure on $N$, what will happen with that in $M'$? Can I say that it still will be from the regular level set theorem? $\endgroup$ – ejk Oct 1 '20 at 19:29

The key fact if you need, if you want to show that $f\circ F$ is a topological embedding is, that restrictions of continous maps are continous:

If $f:X\to Y$ is a continous map between topological spaces, $A\subseteq X$, $B\subseteq f(A)$, then the restriction $f :A\to B$ is also continous, where $A$ and $B$ are equipped with the subspace topologies.

From this it follows that if $f:X\to Y$ is a homeomorphism and $A\subseteq X$, the restriction $f:A\to f(A)$ is also a homeomorphism.

Now in your case you want to show that $f\circ F:N\to M'$ is a topological embedding, which means that the restriction $f\circ F:N\to f(F(N))$ is a homemorphism.

But this map can be written as the composition

$$N\stackrel{F}\to F(N)\stackrel{f}\to f(F(N))$$

which by the above and the assumption that $F:N\to M$ is a topological embedding is a composition of homeomorphisms and hence a homeomorphism.


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