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I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF.

I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (such as hyperbolic surfaces, Teichmuller theory). I have a background in 2-dimensional hyperbolic geometry (S. Katok, Beardon), and 3-dimensional hyperbolic geometry (basic stage), (basic) algebraic topology (Munkres), Riemann surfaces, smooth manifolds, (basic) Riemannian geometry. I want to study more on hyperbolic geometry towards the following direction: the relation between Anti-de Sitter geometry and Teichmuller theory (see here). I want to learn Anti-de sitter geometry.

I found that Anti-de sitter geometry intersects (may be a subtopic) with Lorentzian $(n; 1)$ geometry. Moreover, one can examine the constant curvature spaces: flat, de Sitter (positive curvature) and anti-de Sitter (negative curvature) spaces in context of Lorentzian geometry. So, I think anti-de Sitter might be similar type of hyperbolic geometry. Also, I have heard that (flat) Lorentz $(3; 1)$ space is the natural home for Einstein's Special Theory of Relativity, with three "space" dimensions and one "time" dimension. But, I am not a student from physics background. I want to study the hyperbolic geometric aspects of the subject. I am eager to learn Lorentzian geometry and anti-de Sitter geometry thoroughly and rigorously. Also, I want to learn its relations with Teichmuller theory (see here, Prop. 22 for the Earthquake Theorem).

I did not find any books related to the above said topics e.g. Lorentzian geometry and anti-de Sitter geometry. I scouted the internet and found some papers on the topics such as "Crooked surfaces and anti-de Sitter geometry" (see here), "The geometry of crooked planes" (see here), "Fundamental polyhedra for Margulis space-times" (see here), "The Margulis Invariant of Isometric Actions on Minkowski $(2+1)$-Space" (see here), etc. But I do not know about the proper way to study the subject. Also, I found that those papers discussing Lorentzian geometry (anti-de Sitter geometry, in particular ) also discuss about Einstein universe, crooked planes, affine transformations, proper affine actions, Margulis space-times, Margulis Invariant, crystallographic groups, etc. I have no idea about the previous said keywords, e.g. crooked planes, Margulis space-times. I want to know why one should discuss crooked planes, Margulis space times to learn Lorentzian geometry (in particular, anti-de Sitter geometry).

I want to learn Lorentzian geometry and anti-de Sitter geometry thoroughly and rigorously. Then I want to focus on its relation with Teichmuller theory (may be later).

Please advise me about a learning roadmap of Lorentzian geometry and anti-de Sitter geometry.

Sorry for my bad English. Please help me.

Thanking you in advance.

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2 Answers 2

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One possibility is to start with some overview article by experts in the field. Most likely, you will not be able to understand everything from the beginning. You should then consult the references and start reading details of interesting aspects. Here is a possible article, which is not really an overview article but an open problem list, but it seems to be a good starting point (https://arxiv.org/pdf/1205.6103.pdf).

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A good starting point for the topics you want to study, considering your stated background, is the book by Barrett O'Neill, Semi-Riemannian Geometry (Academic Press, 1983), especially Chapter 4. That should provide adequate preparation for e.g. the review article by Barbot et alii quoted in Sebastian's answer. A more extensive list of references can be found at this math.SE question.

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  • $\begingroup$ The ch. 4 of the book by Barrett O'Neill, Semi-Riemannian Geometry and MSE question give an introduction toward general relativity. But, I am more interested in learning the hyperbolic geometric aspects of the subject, namely, its relation with Hyperbolic manifolds and Teichmuller theory. $\endgroup$
    – user2022
    May 26, 2021 at 15:29
  • $\begingroup$ The behavior of constant curvature in Lorentz signature has some resemblances with the Riemannian case, but also many differences. The behavior of timelike geodesics, for instance, is reversed - de Sitter timelike geodesics drift away exponentially fast from each other (as they would in hyperbolic space), whereas anti-de Sitter timelike geodesics tend to refocus back (as they would in a sphere). O'Neill's book explains these differences beautifully and thoroughly. Moreover, any book on Lorentzian geometry is bound to relate to GR, as it's both its source and main field of applications. $\endgroup$ May 26, 2021 at 15:54
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    $\begingroup$ O'Neill's reference is really meant as a starting point, especially considering your stated background, as I said above. Before delving into anti-de Sitter geometry, one needs to be acquainted with Lorentzian geometry proper, specially regarding what makes it different from Riemannian geometry. O'Neill's book is as good a place for that as any. $\endgroup$ May 26, 2021 at 15:59

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