Timeline for Reconstructing a Lie group from its Maurer-Cartan form (role of completeness)
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9 at 21:37 | comment | added | Alex Bogatskiy | amazing, thanks Ben. | |
| Jul 9 at 12:15 | comment | added | Ben McKay | The latest draft of my Introduction to Cartan geometries (arxiv.org/abs/2302.14457) gives the proof I gave above, with much more detail. | |
| Jul 8 at 22:22 | comment | added | Alex Bogatskiy | ah, of course. Thank you! I have yet to understand developments in more general Cartan geometries, so this is a very nice prototype. | |
| Jul 8 at 17:56 | comment | added | Ben McKay | By the Frobenius theorem, the developments lie on the leaves. We can look at the developments as just lifting each curve in $M$ by the covering map $\tilde{M}\to M$, then mapping each point of that lifted curve by $\tilde{M}\to G$. | |
| Jul 8 at 17:46 | comment | added | Ben McKay | @AlexBogatskiy: I agree that the proof by developments is valuable. I am ashamed that even after reading Sharpe's book (when I was his master's student), I wrote a paper proving Ehresmann's theorem, without noticing that it was already known. | |
| Jul 8 at 16:01 | vote | accept | Alex Bogatskiy | ||
| Jul 8 at 16:01 | comment | added | Alex Bogatskiy | That's a cool proof, although I'm a bit perplexed as to why Sharpe didn't include it (the Ehresmann theorem is already in the book!). Nevertheless, the proof via developments also seems quite valuable. | |
| Jul 6 at 9:19 | history | answered | Ben McKay | CC BY-SA 4.0 |