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Lets start off with the definition of diffeomorphism from Wolfram MathWorld:

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.A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

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