Timeline for Which vector bundle are the Christoffel symbols sections of?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2012 at 13:45 | comment | added | Paul Siegel | If $\nabla$ is a connection on a vector bundle $V \to M$ then one often writes in local coordinates $\nabla_{\partial_i} = \partial_i + \Gamma_i$ where $\Gamma_i$ is a section of the bundle $End(V)$. Regarded as a $n \times n$ matrix, the $(j,k$ entry of $\Gamma_i$ is the Christoffel symbol $\Gamma_{ij}^k$. I don't think you can do better than that because $\nabla_X$ does not depend $C^\infty(M)$-linearly on $X$. | |
| Feb 4, 2012 at 13:42 | answer | added | BS. | timeline score: 9 | |
| Feb 4, 2012 at 13:33 | answer | added | Robert Bryant | timeline score: 15 | |
| Feb 4, 2012 at 13:32 | comment | added | Deane Yang | Yes, it's naturally a section of a vector bundle over not $M$ but over the frame bundle (the bundle of, say, orthonormal bases of $T_xM$) of $M$. It's not possible that it be a section of a vector bundle over $M$, because the Christoffel symbols depend on not just a point on $M$ but also a choice of co-ordinates (or at least an infinitesimal choice which is what a frame of tangent vectors effectively is). | |
| Feb 4, 2012 at 13:10 | history | asked | Qfwfq | CC BY-SA 3.0 |