Timeline for Manifolds admitting flat connections
Current License: CC BY-SA 3.0
10 events
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| Oct 14, 2014 at 2:58 | history | edited | Marcos Cossarini | CC BY-SA 3.0 |
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| Oct 14, 2014 at 2:49 | history | edited | Marcos Cossarini | CC BY-SA 3.0 |
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| Oct 14, 2014 at 2:42 | history | edited | Marcos Cossarini | CC BY-SA 3.0 |
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| Oct 13, 2014 at 17:51 | comment | added | Vladimir S Matveev | It is almost an offtopic but the statement that the n-tori are the only closed n-manifolds that admit a locally free $R^n$ action is standard in the theory of integrable systems and a good proof can be found in the Arnold's ``Mathematical methods of classical mechanics''. There are of course proofs in other sources, say in Berger's ''Geometry''. Though the statement is intuitively expected, the proof is still slightly tricky and may require some time. | |
| Oct 13, 2014 at 11:37 | comment | added | Marcos Cossarini | I corrected the "symmetric flat" thing. I like the idea of using Cartan-Hadamard to prove that spheres don't admit torsionfree flat connections. This saves the task of proving that only toruses admit free $\mathbb R^n$ actions, which is something I didn't do in my answer. | |
| Oct 13, 2014 at 11:32 | history | edited | Marcos Cossarini | CC BY-SA 3.0 |
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| Oct 12, 2014 at 7:49 | comment | added | Vladimir S Matveev | Indeed, you explained that for flat torsionsfree connection on a simply connected n-manifold there exist n linearly independent parallel vector fields. Take a Riemannian metric such that at every tangent space in the basis given by these parallel vector fields it is standard euklidean. Since it is constructed by parallel objects, it is also parallel which implies that our (flat torsionfree) connection is its Levi-Civita connection. Then, the metric is flat and the argument you used at the end of your answer (with Cartan-Hadamard) can be used also here to show that it is impossible | |
| Oct 12, 2014 at 7:47 | comment | added | Vladimir S Matveev | P.S. I have edited my answer a bit without reading yours first, sorry for it, and now it may be an overlap: in the current version of my answer I also claim that any sphere does not admit a flat symmetric connection. Actually, your explanation at the beginning of your answer is (almost) an explanation why any sphere does not admits a flat torsionfree connection, I will explain it in the next comment | |
| Oct 12, 2014 at 7:46 | comment | added | Vladimir S Matveev | What do you mean by symmetric connection in your answer concernings Question 3? Is it torsionfree? If yes, may be you should slightly correct your answer by saying symmetric flat instead of symmetric? | |
| Oct 12, 2014 at 4:25 | history | answered | Marcos Cossarini | CC BY-SA 3.0 |