Timeline for Is there a necessary and sufficient condition for the tangent bundle of a fiber bundle to be trivial?
Current License: CC BY-SA 2.5
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| when toggle format | what | by | license | comment | |
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| Apr 12, 2010 at 16:08 | vote | accept | krolik | ||
| Apr 10, 2010 at 14:24 | comment | added | Deane Yang | I really like this answer; it gives a nice generalization to the fact that the tangent bundle of a Lie group is trivial. Indeed, the tangent bundle of the principal $O(n)$-bundle of orthonormal frames of a Riemannian manifold is parallelizable. This is a standard fact in exterior differential systems, explained often by Robert Bryant. It seems to me that the same argument shows that any principal $G$-bundle constructed from tangent frames (not necessarily orthonormal) has a parallelizable tangent bundle. You just need to construct a connection on that bundle to get the trivialization. | |
| Apr 10, 2010 at 8:19 | history | answered | Benoît Kloeckner | CC BY-SA 2.5 |