I want to find a local diffeomorphism $\Bbb{R}^2\to\Bbb{R}^2$ that is not a diffeomorphism onto its image. This is what I thought: $f(x,y)=(\sin 2\pi x, \cos 2\pi y)$. Does that work? Seems ok to me.
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1What does the derivative look like that the point $(1/4, 0)$? – froggie Feb 23 '14 at 11:24
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1Yeah, the jacobian is not invertible so it is not a local diffeomorphism at this point. thanks – helix Feb 23 '14 at 11:29
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2the exponential function does it quite nicely (on $\mathbb{C}$ which is isomorphic to $\mathbb{R}^{2}$) – Max Feb 23 '14 at 11:33
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http://en.wikipedia.org/wiki/Exponential_function#Complex_plane – Max Feb 23 '14 at 11:41
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possible duplicate of surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism – Moishe Kohan Feb 24 '14 at 06:36
