Diffeomorphism preserves dimension?

Question

I want to apply the inverse function theorem to a general mapping that is a diffeomorphism. The inverse function theorem, as stated in my textbook, requires that the dimension of the range and domain spaces be equal. I see in the post Diffeomorphism preserves dimension

that this is indeed the case where the range and domain of the mapping are open subsets of $\Bbb R^n$ and $\Bbb R^m$. My question is, can you have a diffeomorphism between all of $\Bbb R^n$ and all of $\Bbb R^m$ where $n$ and $m$ are not equal?

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score 4 · Answer 1 · answered Jul 14 '18 at 06:58

No, since a diffeomorphism $f$ between smooth manifolds $M$ and $N$ induces a vector space isomorphism between tangent spaces $T_p(M)$ and $T_{f(p)}(N)$ for all $p\in M$. This means the tangent spaces have the same dimension, and so the manifolds have the same dimension.

This is also true for homeomorphisms, but the proof is harder (see Brouwer's "invariance of domain" theorem).

Diffeomorphism preserves dimension?

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