Timeline for Reconstructing a Lie group from its Maurer-Cartan form (role of completeness)
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jul 8 at 16:01 | vote | accept | Alex Bogatskiy | ||
| Jul 6 at 9:19 | answer | added | Ben McKay | timeline score: 4 | |
| Jun 29 at 18:26 | comment | added | Alex Bogatskiy | @DmitriPavlov Got it, thanks a lot! | |
| Jun 29 at 17:37 | comment | added | Dmitri Pavlov | These are local liftings, whose existence is established by (the generalization of) Theorem 6.1. Alternatively, you can use local liftings to define a local exponential map, like you proposed. Both choices work fine. | |
| Jun 29 at 16:58 | comment | added | Alex Bogatskiy | when you say "lift to $M$", are you using some local diffeomorphism between $G$ and $M$ given by something like $\exp\circ\omega$? I guess I'm wondering why I can't do without $G$. I think I can define a local exponential map on $M$ and use that to deform paths to turn them into piecewise integral curves of constant vector fields via a flat homotopy. Wouldn't that be enough? | |
| Jun 29 at 14:03 | comment | added | Dmitri Pavlov | What is being approximated is the differential 1-form on [a,b] with values in the Lie algebra, which does not involve M. The manifold M gets involved only at the last stage, when we choose ε>0 so that the developed homotopy stays inside M. The developed homotopy always exists locally on [a,b] (as already proved in the book), and the global extension (for sufficiently small ε>0) is possible because [a,b] is compact. | |
| Jun 29 at 7:05 | comment | added | Alex Bogatskiy | @DmitriPavlov it seems that we need the proof of the existence of piecewise-constant approximations to also hold on M. That should work, because the analog of the exponential map exists here too, and is a diffeomorphism by the same argument as in Lie groups. | |
| Jun 29 at 4:35 | comment | added | Dmitri Pavlov | G is a Lie group (by definition). The manifold M is not a Lie group (yet). | |
| Jun 29 at 3:07 | comment | added | Dmitri Pavlov | By making the subdivision sufficiently small, we can ensure the homotopy moves points at most by some ε>0. Now working with an arbitrary manifold M equipped with a Lie-algebra-valued 1-form ω such that dω+[ω∧ω]=0, using compactness of the interval [a,b], we ensure the development of the piecewise-constant approximation to ω exists in M (by assumption), and, furthermore, the flat homotopy lifts to M (here we make ε>0 small enough). Therefore, the development of the original path exists. | |
| Jun 29 at 3:05 | comment | added | Dmitri Pavlov | A quick and dirty proof of the existence of flows for nonconstant vector fields can be given as follows. Since the exponential map is a local diffeomorphism in any Lie group G, every smooth path in G can be homotoped to a piecewise smooth path whose Darboux derivative is piecewise constant. Therefore, every Lie algebra-valued differential 1-form can be homotoped to a piecewise constant Lie algebra-valued differential 1-form via a flat homotopy (i.e., with a vanishing Maurer–Cartan form). | |
| Jun 29 at 2:40 | comment | added | Alex Bogatskiy | @DmitriPavlov oh, is the argument for developments simply that $\omega^{-1}(\eta(\dot\gamma(t)))$ is a time-dependent vector field, which at every $t$ is complete, and hence is complete itself? I'm not sure if such a theorem for flows of time-dependent vector fields actually holds though. | |
| Jun 28 at 23:40 | history | edited | Alex Bogatskiy | CC BY-SA 4.0 |
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| Jun 28 at 22:59 | comment | added | Alex Bogatskiy | @DmitriPavlov do you know how to show the existence of a unique primitive? On Lie groups, I can just take a finite number of local primitives and stitch them together by translating their values. But without a group structure it seems like I would need to show that any automorphism of $(G,\omega)$ is uniquely determined by its value at one point. And I'm not sure how to do that. | |
| Jun 28 at 19:20 | comment | added | Dmitri Pavlov | You correctly pointed out that the fundamental theorem should be generalized in the way you described. Completeness is necessary to prove the analogous generalization of Theorem 7.1. The second paragraph of its proof implicitly uses the fact that there is a unique primitive taking the given value at some fixed point in the interval. The analogous statement for arbitrary M uses the completeness property of the 1-form precisely at this point. | |
| Jun 28 at 18:32 | comment | added | Alex Bogatskiy | @DmitriPavlov ah, indeed, just foreword. | |
| Jun 28 at 18:23 | comment | added | Dmitri Pavlov | Sharpe is the only author of the cited book. Chern is not an author. | |
| Jun 28 at 14:36 | history | edited | Alex Bogatskiy | CC BY-SA 4.0 |
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| Jun 28 at 14:28 | history | asked | Alex Bogatskiy | CC BY-SA 4.0 |