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Jun 8, 2023 at 8:47 comment added jens I couldn't find an accessible proof. It seems clear to me that your function is proper since compact sets in the codomain have compact preimages (nothing escapes to infinity). So that shouldn't be the issue. I suspect, however, that your counterexample is not smooth at the transition between both parts of the curve, at $t=2\pi$. Could that be the issue? On the other hand, the existence of the bump function (see en.wikipedia.org/wiki/Smoothness) gives me some confidence that your counterexample could be made smooth. So, I am still puzzled.
Jun 7, 2023 at 17:46 comment added Iosif Pinelis I am not familiar with the proof of that theorem. I would guess that the map in my example is not proper in the sense used in that theorem. You are welcome to check that.
Jun 7, 2023 at 17:23 comment added jens May I ask what in your counterexample violates the assumptions for the Hadamard-Cacciappoli Theorem as stated in en.wikipedia.org/wiki/Diffeomorphism? I can't figure out.
Sep 6, 2020 at 16:50 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 16:40 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 12:55 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 12:49 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 12:44 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 12:20 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 12:20 comment added Piotr Hajlasz This is a very nice example.
Sep 6, 2020 at 10:47 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 4:21 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 3:31 history edited Iosif Pinelis CC BY-SA 4.0
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Sep 6, 2020 at 3:10 history answered Iosif Pinelis CC BY-SA 4.0