Need help with planning self study for learning differential/Riemannian geometry and General Relativity rigorously.

differential-geometrygeneral-relativityreference-requestriemannian-geometrysoft-question

I would like to learn Mathematics for understanding GR, Differential Geometry, Riemannian Geometry and related research papers rigorously.

I would like to carve out a clear path to understand these topics by listing out all the necessary prerequisites.

I have undergrad Math under my belt such as: Real Analysis, Algebra, Topology and ODEs. I am missing intro to PDEs at this point.

I have also created a diagram of the prerequisites in which each bubble represents a subject along with textbooks written in blue.

Please take a look at the attached image/file. UG means "Undergrad" in the diagram.

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Specifically, I need help with the following questions:

  1. Is my goal (the center bubble) well defined? I know it may not be specific enough yet, but I have tried to list down some topics I am interested in RED color.
  2. Have I listed all the subjects? Am I missing any subject?
  3. Is Lie Groups worthy of mention here? Or, would it just fit under Algebra?
  4. Is Hyperbolic Geometry worthy of mention here? Is it relevant? How do I learn it? Any textbooks for it?
  5. Would anyone please help me break down the following subjects into specific topics that are necessary for my goal: Manifolds, Riemannian Geometry, Real Analysis (grad version), PDEs (grad version), Algebra (grad version).

Best Answer

I suggest reading

O’Neill, Barrett, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. xiii, 468 p. (1983). ZBL0531.53051.

It is 100% rigorous and covers most of what you want to learn, including basics of Lie groups, smooth manifolds and Riemannian geometry. Afterwards, you will likely need to learn hyperbolic geometry, Lie groups and their representations in greater detail, as well as general principal bundles and vector bundles, and connections/curvature on general bundles.

Two more references:

Hall, Brian, Lie groups, Lie algebras, and representations. An elementary introduction, Graduate Texts in Mathematics 222. Cham: Springer (ISBN 978-3-319-13466-6/hbk; 978-3-319-13467-3/ebook). xiii, 449 p. (2015). ZBL1316.22001.

Chapter 1 ("Geometry") of

Jost, Jürgen, Geometry and physics, Berlin: Springer (ISBN 978-3-642-00540-4/hbk; 978-3-642-00541-1/ebook). xiv, 217 p. (2009). ZBL1176.53001.

However, you need a real advisor to guide you.

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