Timeline for connections and curvature
Current License: CC BY-SA 3.0
11 events
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| Aug 13, 2014 at 1:20 | comment | added | Robert Bryant | @StevenGubkin: Actually, just because the problem has many solutions in the degenerate case that the curvature vanishes, that doesn't mean that it has many solutions in the 'generic' case. In fact, most of the time, in dimension $3$ or more, the Riemann curvature tensor completely determines the metric and, for most affine connections in dimension $4$ or more, knowing the curvature tensor determines the connection completely. It's only in 'degenerate' cases or in the low dimensions that the curvature doesn't determine the connection. | |
| Aug 13, 2014 at 1:14 | comment | added | Robert Bryant | @PaulSiegel: While what you say is correct, it doesn't have much bearing on the OP's question. Just because two metrics on a surface have the same Gauss curvature (a function), that does not imply that they have the same (Riemann) curvature tensor (which has type $(1,3)$, as opposed to the Gauss curvature, which is a function). In fact, two metrics on a surface that have the same (nonvanishing) curvature tensor and the same Gauss curvature are equal. | |
| Aug 12, 2014 at 22:35 | comment | added | 314159. | thank you very much for the link and the description given there! | |
| Aug 12, 2014 at 21:25 | comment | added | Robert Bryant | Probably, your question would be better posed as: Given a potential curvature tensor (i.e., 2-form with values in endomorphisms of the tangent bundle), when can it be written as the curvature of a connection and in how many ways? A relevant question (and answer) can be found at mathoverflow.net/questions/73439/… | |
| Aug 12, 2014 at 21:18 | comment | added | Paul Siegel | Fix a topological surface of genus $g > 1$. According to Teichmuller theory, there is a $3g - 3$-dimensional space of conformally inequivalent metrics with constant curvature $-1$. | |
| Aug 12, 2014 at 21:05 | comment | added | Vít Tuček | Can we describe the solution space explicitly? | |
| Aug 12, 2014 at 21:03 | comment | added | Steven Gubkin | @314159. Paul answered your question. It should show you that your problem should have many solutions in whatever case you care about. | |
| Aug 12, 2014 at 20:53 | history | edited | 314159. | CC BY-SA 3.0 |
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| Aug 12, 2014 at 20:28 | comment | added | 314159. | thank you for your nice comment!I am interested however more in the not-flat case, I mean $R(X, Y)Z=R'(X, Y)Z\neq 0$. | |
| Aug 12, 2014 at 20:22 | comment | added | Paul Siegel | Yes: take two flat tori with different volumes. | |
| Aug 12, 2014 at 19:56 | history | asked | 314159. | CC BY-SA 3.0 |