Maurer-Cartan structure equation derivation

Dear all.

I'm a theoretical physicist trying to understand the structure equations and their geometrical significance, this for their gravitational applications.

I know the relation between the Lie algebras and the Maurer-Cartan structure equations, but as far as I know, they are used in manifolds which are not necessarily Lie groups.

Questions:

• What are the axioms or conditions on which the structure equations lie on?
• Where can I find a derivation of them?
• Are they valid for any manifold?
-
Can you give a more specific example? As far as I know. Maurer-Cartan structure equations do involve some Lie group, in one form or another. –  Liviu Nicolaescu Jul 27 '12 at 14:52
I suspect he's thinking about the Maurer-Cartan equation that arises in deformation theory a la DGLAs with Kodaira-Spencer theory being the usual (but not only) example that arises in theoretical physics. –  Aaron Bergman Jul 28 '12 at 21:52

What you are asking for is an introduction to the theory of $G$-structures, for which the (Maurer-Cartan) structure equations are a basic tool.

There are many sources for this material, starting, of course, with the fundamental works of Élie Cartan on the subject, though many find his expositors, who use more modern language, easier to follow.

S.-S. Chern has an article, The geometry of G-structures. Bull. Amer. Math. Soc. 72 (1966) 167–219, that I highly recommend as an introduction. There is Sternberg's book, Lectures on differential geometry, which is a good reference towards the end of the book. A more modern treatment specifically on the use of structure equations in modern differential geometry is R. Sharpe's book Differential geometry. Cartan's generalization of Klein's Erlangen program. Graduate Texts in Mathematics 166. Springer-Verlag, New York, 1997. There is also the book by Ivey and Landsberg Cartan for beginners: differential geometry via moving frames and exterior differential systems. Graduate Studies in Mathematics 61. AMS, Providence, RI, 2003.

That's just a small sample, but it should be enough to give you a choice of authors so that you can figure out which comes closest to being able to help you.

-
Thank you for the response. I'll check some of the reference you have recommended. By the way! I'm a big fan of your Park City lectures!! Cheers. –  Dox Jul 27 '12 at 15:24
@Dox: You are most welcome and most kind. I'm glad that you have found the Park City lectures useful. It was the first year of PCMI (1991) and I was staying up nights writing the lectures on an SE-30 just before giving them, so it was quite an experience for me. I've often thought that I would revise them a bit and add a few more topics (such as the convexity theorem, etc.) to bring them more up-to-date, but finding the time has been a problem. –  Robert Bryant Jul 27 '12 at 21:33

To elaborate on Robert's answer a little, the standard Maurer-Cartan equations are for a Lie group only. In particular, they express the structure equations of the associated Lie algebra (i.e., left-invariant vector fields) in terms of the dual Lie algebra (i.e., left-invariant differential forms) and the exterior derivative.

However, extensions of these equations are extremely useful in more general settings such as a Riemannian manifold. You can study a Riemannian metric, its Levi-Civita connection, and its curvature using its orthonormal frame bundle. You could try to do this using vector fields but, as far as I know, this turns into a bit of a mess. What's better is to use differential forms that behave nicely along each fiber. The Maurer-Cartan equations play a natural role here, leading to an elegant way to define and work with the connection and curvature of a Riemannian manifold.

I don't know if Robert has any lecture notes on this but if he does, that would be by far the best exposition available. Of course, if you're lucky enough to hear him lecture on it, that's even better. That's how I learned all of it. But otherwise I would recommend looking at the book by Ivey and Landsburg.

-
Thank you so much for the details, and the reference. I'll look at that book! Cheers. –  Dox Jul 27 '12 at 15:30