Timeline for Constructing jet bundles from a cocycle of smooth transition functions
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2016 at 8:45 | comment | added | Igor Khavkine | @AndreasCap <- this is the right way to ping someone. | |
| Oct 27, 2016 at 7:19 | comment | added | ಠ_ಠ | I'm a bit confused right now. How would you write the cocycles, Andreas Cap? I'd be happy to upvote if you wouldn't mind posting an answer as well. | |
| Oct 21, 2016 at 6:59 | comment | added | Andreas Cap | I see, so you get the right bundle but not in a functorial way, right? | |
| Oct 20, 2016 at 10:12 | comment | added | Igor Khavkine | You are right that the construction above doesn't give the right transition functions for higher order frame bundles, if one wants to construct them as jet prolongations of the first order frame bundle, when the latter is considered as a $GL(n)$ principal-bundle. But I would also argue that it is not the right way to get the higher order frame bundle. In the intermediate bundles $S\times U_i$ above, implicitly, diffeomorphisms of the base act trivially on the $S$-fibers. This is already not true for the frame bundle. | |
| Oct 20, 2016 at 8:33 | comment | added | Andreas Cap | I am afraid that this is not correct in general. The fact that a bundle $E$ is associated to a principal $G$-bundle $P$ does not imply that the $k$th jet prolongation is associated to $P$. Just think of the case where $P$ is the first order frame bundle of $M$, and $E$ is any natural vector bundle. If $J^kE$ were associated to $P$ again, then any diffeomorphsim of $M$, which coincides with the identity to first order in a point $x\in M$ would act trivially on the fiber of $J^kE$ over $x$. | |
| Oct 20, 2016 at 7:39 | vote | accept | ಠ_ಠ | ||
| Oct 20, 2016 at 7:24 | history | answered | Igor Khavkine | CC BY-SA 3.0 |