Timeline for What is the geometric significance of Cartan's structure equations?
Current License: CC BY-SA 2.5
10 events
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| Nov 4, 2009 at 2:52 | comment | added | Akhil Mathew | Thanks again for the links! The interpretation of Christoffel symbols as components of a 1-form is cleaner than the usual one. Now I am going to take a look at the others. | |
| Nov 4, 2009 at 2:31 | vote | accept | Akhil Mathew | ||
| Nov 3, 2009 at 8:50 | comment | added | Urs Schreiber | I have now some discussion of how Christoffel symbols relate to the general notion of connection here ncatlab.org/nlab/show/Christoffel+symbol . That should connect from what you are thinking about to the description of Lie-algebra valued 1-forms that I gave. Much more can be said here, but it's a start and I have to do something else now for while. | |
| Nov 3, 2009 at 7:44 | comment | added | Urs Schreiber | Ah, I see. Well, no problem. To see how everything I am saying here pertains to "Christoffel symbols" you just need to know how a "Christoffel symbol" is a connection 1-form on the tangent bundle written out in a fixed basis. I'll say more about that. For the moment, I have typed the details of the geometric interpretation of the curvature 2-form (whic is possibly part of what you were really asking for originally). See here ncatlab.org/schreiber/show/… | |
| Nov 3, 2009 at 2:34 | comment | added | Akhil Mathew | Hm, actually I was referring precisely to the Christoffel equation formulation of the Cartan equations. I did not know that something else existed. I am vaguely aware that a g-valued 1-form determines the connection (but this is the Ehresmann definition, not the Koszul one, right?) though I probably should read up more on moving frames about your assertion on so(n). Anyway, thanks for the links! I'm going to take a look. | |
| Nov 3, 2009 at 1:36 | comment | added | Urs Schreiber | Okay, I have now started typing a section "ordinary Lie-algebra valued 1-forms" into the general context here: ncatlab.org/schreiber/show/… . There is more to be said, but I have to stop now, as it is late at night here. More tomorrow or (since I'll be very busy the next two days) on Thursday. Let me know what you think and I can go into more details on whatever you desire. (Possibly what I said so far doesn't answer anything that you didn't already know. So let me know what you need.) | |
| Nov 3, 2009 at 0:49 | comment | added | Urs Schreiber | Ah, one comment: I realize that among some people "Cartan's structural equations" are thought of as some property of -- ahem -- Christoffel symbols. (e.g. on the otherwise usually quite good PlanetMath.) May I assume that you are familiar with how connections are locally given by Lie-algebra valued 1-forms and how "Christoffel symbols" are the expression of a nice so(n)-valued 1-form in a funny basis? | |
| Nov 3, 2009 at 0:38 | comment | added | Urs Schreiber | Right, as I said, you should make me spell out the full general nonsense in nice bite-sized examples. I'll type one for you now, but will probably have to call it quits soon, as it is getting later here. Most of what I am going to say is in one way or other spelled out as an example in my article with Stasheff and Sati arxiv.org/abs/0801.3480 , though that too may look more overwhelming than this topic really is. Give me 15 minutes and I'll get back to you with the simplest interesting example. | |
| Nov 3, 2009 at 0:20 | comment | added | Akhil Mathew | Wow. This certainly looks general, though unfortunately I'm still struggling to keep my n category number at about 2. | |
| Nov 3, 2009 at 0:04 | history | answered | Urs Schreiber | CC BY-SA 2.5 |