The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations- can one visualize the proof?
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asked Nov 2, 2009 at 20:33
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2 Answers
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There is a nice grand story behind all this. I don't know if you like thinking that way, but things do clarify when one looks at it from a more general perspective of oo-Lie algebroid valued differential forms with curvature.
I am still working on these entries, trying to expose some of my work with Jim Stasheff and Hisham Sati. If you bug me with questions, there is a good chance that I'll improve the exposition taylor-made for your needs.
answered Nov 3, 2009 at 0:04
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2$\begingroup$ Wow. This certainly looks general, though unfortunately I'm still struggling to keep my n category number at about 2. $\endgroup$ Commented Nov 3, 2009 at 0:20
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$\begingroup$ 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. $\endgroup$ Commented Nov 3, 2009 at 0:38
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2$\begingroup$ 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.) $\endgroup$ Commented Nov 3, 2009 at 1:36
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1$\begingroup$ 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/… $\endgroup$ Commented Nov 3, 2009 at 7:44
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1$\begingroup$ 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. $\endgroup$ Commented Nov 3, 2009 at 8:50
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Well, this is how I'd think about getting the curvature from a 1-form, but this might not be what you were asking about.
Suppose we have a connection on a G-bundle. Curvature measures what happens when we transport around a small loop. We can take this to be a parallelogram: move in direction X, direction Y, then direction -X, then direction -Y.
Now, once we have fixed a trivialization of the G-bundle at every points, the effect of moving in direction X is described by an element gX in G, close to the identity. The effect of the parallelogram motion is g_X g_Y g_X^{-1} g_Y^{-1}, but corrected slightly (because the effect of moving along edge -X is not the inverse of moving along X; the two edges are displaced.)
The former term corresponds in usual notation to [\omega, \omega]: it measures the noncommutativity of G. The "correction" corresponds to d\omega: it measures the variation in the 1-form across the parallelogram. Hope this helps.
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$\begingroup$ Have you thought about asking the administrators to merge your many accounts with the same name? $\endgroup$– S. Carnahan ♦Commented Nov 3, 2009 at 3:28
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$\begingroup$ I have typed out details of this argument here: ncatlab.org/schreiber/show/… $\endgroup$ Commented Nov 3, 2009 at 7:45