How do I show that if there is a diffeomorphism between two smooth manifolds, the manifolds have the same dimension?

Question

I think I've somehow got to bring tangent spaces into the picture, but how do I solve this?

What have you done so far? – Robin Goodfellow Oct 08 '14 at 14:26 — Robin Goodfellow, Oct 08 '14 at 14:26

score 3 · Accepted Answer · answered Oct 08 '14 at 14:23

3

The dimension of a manifold is the same as the dimension (as a real vector space) of its tangent space. A diffeomorphism between manifolds induces (at each point) an isomorphism of the corresponding tangent spaces.

answered Oct 08 '14 at 14:23

GFR

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How do I show that if there is a diffeomorphism between two smooth manifolds, the manifolds have the same dimension?

1 Answers1