This may be a stupid question, but I have not seen anywhere that it is said that a diffeomorphism must be an immersion and a submersion. Therefore I am asking the following two questions:

(I) Is a diffeomorphism between smooth manifolds necessarily a submersion?

(II) Is a diffeomorphism between smooth manifolds necessarily an immersion?

(III) If either (I) or (II) aren't true, can someone help by giving a counter example?

Thanks in advance.


Since the derivative of a diffeomorphism at every point is bijective, every diffeomorphism is both a submersion and an immersion.

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  • $\begingroup$ Just one further question if I may, are two diffeomorphic smooth manifolds essentially the same object in an appropriate category? $\endgroup$ – Keen-ameteur Jan 30 '19 at 10:07
  • $\begingroup$ Yes: in the category of differentiable manifolds. $\endgroup$ – José Carlos Santos Jan 30 '19 at 10:09

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