Timeline for Global description of the Levi-Civita connection
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2013 at 10:04 | comment | added | hoj201 | Here is a similar post where I provided one coordinate-free answer mathoverflow.net/a/138073/16852 | |
| Jul 10, 2010 at 6:36 | answer | added | Dick Palais | timeline score: 7 | |
| Apr 24, 2010 at 19:16 | comment | added | Dan Ramras | I'm still curious to see a description in terms of jet bundles. Somehow T(TX) is a strange object, with its two different vector bundle structure (and hence two different notions of fiber-wise addition). This really stands out in the description of H I sketched above, and Tim's suggestion seems to keep this issue more hidden. I guess I haven't thought about it enough to guess whether the jet bundles approach might make any of this structure clearer. | |
| Apr 24, 2010 at 19:15 | comment | added | Dan Ramras | Okay, I think I'm beginning to understand Tim's first comment. Roughly speaking, this seems to be similar to Lang's discussion of parallel transport with respect to a spray, and I guess this does indeed give the section H I'm looking for. So Tim's comment is really an answer. Thanks! This is very much in line with the way I think about connections on principal bundles, so I'm happy to see the analogy. | |
| Apr 24, 2010 at 18:41 | comment | added | Dan Ramras | Tim and Joel: Lang does explain how to obtain the L-C covariant derivative associated to a Riemannian metric on a Hilbert manifold, and he gives the global Kozul formula. So Tim is right that this works on Hilbert manifolds. (It does sound, from Lang's book, that there's an issue on general Banach manifolds: there metrics don't make sense, but the correspondence between sprays and covariant derivatives breaks down.) | |
| Apr 24, 2010 at 14:23 | comment | added | Tim Perutz | (I mean IVPs for 1st order linear ODE in Hilbert space.) | |
| Apr 24, 2010 at 13:56 | comment | added | Tim Perutz | Joel, point taken. I think one can solve initial-value problems in Hilbert space (by finite-dimensional approximation), so maybe we're OK in Hilbert manifolds? But now I see why one might look to jet bundles and the like. | |
| Apr 24, 2010 at 7:59 | comment | added | Joel Fine | @Tim, perhaps the problem is that you can't necessarily find a horizontal section of $\gamma^*TX$ with arbitrary initial condition. In finite dimensions this always works because you're solving an ODE, but in infinite dimensions this becomes a PDE. I remember reading somewhere that the Koszul formula didn't necessarily imply existence of the Levi-Civita connection in infinite dims, but this was a throw-away remark and I never thought about it again until now... | |
| Apr 24, 2010 at 2:56 | comment | added | Tim Perutz | Dan, why not go via the differential operator description? In finite dimensions, if $nabla$ is a covariant derivative in $TX$, the horizontal lifts of a vector in $X$ tangent to a curve $\gamma$ are the sections of $\gamma^* TX$ in the kernel of $\gamma^*\nabla$. Moreover, there's a coordinate-free (Koszul) formula for the L-C covariant derivative. Does something go wrong in infinite dimensions? | |
| Apr 23, 2010 at 23:04 | history | asked | Dan Ramras | CC BY-SA 2.5 |