I'm a physics student trying to understand what exotic manifolds, such as exotic R4, means. Is there known examples what the Riemannian metric of some exotic R4 (or some exotic sphere) would be? Does an exotic R4 have non-zero intrinsic curvature?
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9$\begingroup$ While I applaud your interest, I'm not sure the user name you have chosen is that wise, unless your name happens to be Kirby. Professor Robion Kirby is a well-known low-dimensional topologist who has worked on 4-manifolds! $\endgroup$– David Roberts ♦Commented Sep 16, 2019 at 13:42
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1$\begingroup$ I feel confused with the professor name as well. Maybe there is a physicist named Kirby? $\endgroup$– Bombyx moriCommented Sep 16, 2019 at 14:08
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2$\begingroup$ see also mathoverflow.net/questions/89415/best-metrics-on-exotic-r4 $\endgroup$– Francesco PolizziCommented Sep 16, 2019 at 14:50
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$\begingroup$ As far as I know: it is not known whether an exotic sphere admits a Riemannian (or even Finsler) metric all of whose geodesics are closed. This is a question of Gromoll, if I remember correctly. If a smooth $R^4$ admits a complete metric of zero curvature, then by Cartan-Hadamard it is diffeomorphic to the standard $R^4$. $\endgroup$– alvarezpaivaCommented Sep 17, 2019 at 13:10
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