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Math Help - Ok, I know manifolds. What's next?

  1. #1
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    Ok, I know manifolds. What's next?

    I'm struggling to learn general relativity rigorously. In the process, I've learned some topology (Crossley - Essential Topology, Lee - Introduction to Topological Manifolds) and smooth manifold theory (Tu - An Introduction to Manifolds).

    I now know about manifolds, differential forms, Stoke's theorem, de Rham cohomology and more. But I still haven't got a clue about differential geometry, which GR is really all about.

    Where should I go now?
    How do I learn the geometry stuff in a good way?
    I've heard good things about O'Neill - Semi-Riemannian Geometry, but I'm a little anxious I don't have the geometrical prerequisities. Also, I've been considering Spivak and/or Boothby.

    What would you guys suggest?
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  2. #2
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    General Relativity overlaps mostly with Riemannian Geometry. This is because the theory concerns the metric of space-time. The field equations, for example, is a tensor equation determining the metric. There are many good texts for Riemannian Geometry but you could start with Klingenberg and pick up the prerequisite stuff as you go along. You should also be familiar with tensors and so get a good book on multilinear algebra. Spivak's books are good but there are 5 of them and they cover a lot of stuff that's not immediately relevant. Finally, you need to be relating this all back to physics. I think Misner, Thorne, and Wheeler's book Gravitation gives a good intro to the subject.
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