Timeline for What is the Levi-Civita connection trying to describe?
Current License: CC BY-SA 4.0
8 events
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| Nov 18, 2020 at 8:48 | comment | added | Ben McKay | @BenCrowell: flatness of the ambient metric is not needed; any pseudo-Riemannian metric induces such a metric, on any submanifold on which the metric restricts to be pseudo-Riemannian. | |
| Nov 17, 2020 at 3:16 | comment | added | Deane Yang | @BenCrowell, yes. Any flat Riemannian manifold is locally isometric to Euclidean space. The analogous fact is true for flat semi-Riemannian spaces, too. | |
| Nov 17, 2020 at 2:41 | comment | added | user21349 | "on a submanifold of Euclidean space, the flat connection on Euclidean space naturally induces a connection on the submanifold, and that connection is indeed torsion-free." You say Euclidean, but does this actually hold for a submanifold of any flat space, whether Riemannian or semi-Riemannian? If not, then this seems less attractive to me as a motivation. | |
| Nov 15, 2020 at 5:23 | comment | added | Deane Yang | @user44191, here's an example: Note that since the Riemann curvature tensor is uniquely determined by the metric alone, you can define it without ever using the Levi-Civita connection. This is demonstrated in a little note I wrote, math.nyu.edu/~yangd/papers/riemann.pdf. However, this is all encoded more elegantly using the definition of the Levi-Civita connection and the definition of the curvature tensor using it. | |
| Nov 15, 2020 at 5:00 | comment | added | Deane Yang | @user44191, I can only repeat what I said. If you want to study properties of a Riemannian metric together with a connection, then you can omit the torsion-free assumption and investigate the properties of the pair. If you want to study the properties of only the Riemannian metric itself, you could still study the Riemannian metric with a connection (possibly with torsion) and then identify which of the results do not depend on the connection chosen. It, however, is far easier to use a connection that is uniquely determined by the metric. | |
| Nov 15, 2020 at 4:49 | comment | added | user44191 | I think your answer might work better with (alternatively, I personally would appreciate) an example of the "payoff" that works with a torsion-free connection but not a torsion connection; is there some "gateway" theorem that only works with torsion-free connections, behind which you find the richness of torsion-free geometry? | |
| Nov 15, 2020 at 4:34 | history | edited | Deane Yang | CC BY-SA 4.0 |
added 635 characters in body
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| Nov 15, 2020 at 4:27 | history | answered | Deane Yang | CC BY-SA 4.0 |