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Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've already looked the references given in this answer, but it is not exactly what I'm looking for.

I also took a look in O'Neill's "The Geometry of Kerr Black Holes", but it wasn't helpful to me: the way he defines the connection forms $\omega_{ij}$ is different than the way he defines them in his first book Elementary Differential Geometry (indices swapped, and something more, it seems). I managed to adapt the connection equations, and his expression for the connection forms $\omega = dA \cdot A^t$ (page $91$), with some changes, but I'm failing to get the structural equations (page $95$). My level of background is more or less of this book.

I'm not familiar with tensors. Just more or less with differential forms.

If someone knows any text who does something like that, it would help me a lot. Thanks.

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Ivo Terek
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  • do you mean something like http://planetmath.org/cartanstructuralequations ?? – janmarqz Dec 13 '14 at 01:16
  • More or less. The way I saw the theory so far doesn't use Christoffel symbols, and I haven't studied the curvature tensor $R$ yet.. – Ivo Terek Dec 13 '14 at 01:40
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    maybe in the direction http://en.wikipedia.org/wiki/Riemannian_connection_on_a_surface#Cartan_structural_equations ? – janmarqz Dec 13 '14 at 19:11
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    Yes, that's very much closer! I want something analogous made in a Lorentzian surface. – Ivo Terek Dec 13 '14 at 21:43
  • then check this http://www.math.ucsd.edu/~benchow/250-12-03.pdf – janmarqz Dec 13 '14 at 22:11
  • leaning toward physics it is http://aflb.ensmp.fr/AFLB-324/aflb324m618.pdf with a lot a details – janmarqz Dec 13 '14 at 23:38
  • quote from W.A. Rodriguez: "After that we solved with details some exercises involving different grades of difficult, ranging depending on the readers knowledge from kindergarten, intermediate to advanced levels. In particular we show how to express the derivative ( d) and coderivative (δ) operators asfunctions of operators related to the Levi-Civita or a Riemann-Cartan connection defined on a manifold, namely... " – janmarqz Dec 13 '14 at 23:40

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You might try Sternberg's Curvature in Mathematics and Physics (Sternberg) or Chern's Lectures on Differential Geometry (Chern). Both are very reasonably priced and make extensive use of moving frames.

THW
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