Equivalence of diffeomorphism definitions

  • #1
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Hi

Lets start off with the definition of diffeomorphism from Wolfram MathWorld:
A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

The issue is that I am learning about smooth manifolds, and in the books I've read, the map has to be smooth and have a smooth inverse. Also, the definition above doesn't say that it has to be bijective. However, the Encyclopedia of Math does.

So, are these definitions equivalent? If not, when are they equivalent?
 

Answers and Replies

  • #2
andrewkirk
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the definition above doesn't say that it has to be bijective. However, the Encyclopedia of Math does.
It says the map has an inverse, which means it is injective. In practice the focus will be restricted to the image of the map, so we have a bijection between the domain and the image of the map.

Smoothness is strictly stronger than differentiability. Consider two identical manifolds that are infinite cylinders. I think it should be easy to construct a bijective, differential map between the two that is not smooth. Just take a bijection on ##S^1## that is once but not twice differentiable and extend it parallel to the cylinder's axis.
 
  • #3
mathwonk
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you should get used to the fact that most terms have many different definitions, so the point is not to ask for a definition that will always be valid in every seting, but to remember in every specific context to ask what that author's definition is. In your case e.g. even the word "smooth" has no universal definition: e.g. in Milnor's little book, Topology from the differentiable viewpoint, it means C^infinity, but in most books, e.g. Mo Hirsh's Differential Topology, it means C^1 or even less precisely, C^r with r ≥ 1.
 

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