I want to study basic properties of symmetric spaces.
What is a basic textbook?
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I do not now really elementary text, but for the non-positively curved ones you can have a look at Eberlein's Geometry of nonpositively curved manifolds (Chicago Lectures in Mathematics, University of Chicago Press, 1996). By the way, I do not see why similar answers where stated as comments. Maybe you should repost them as answers so that one can be accepted? |
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My favorite text is chapter 8 in Joseph Wolf's book "Spaces of constant curvature". I have a copy of the 5th edition, published by Publish of Perish, but a 6th edition by AMS Chelsea has recently come out. I don't claim it is an easy read, you need to work a lot on the details, but it gets to the point very efficiently. Perhaps it is fair to say that one can use it as a guide and complement the arguments as needed using the books of Helgason and Loos (2nd volume). In particular, the classification of symmetric spaces is done in a rather elementary way, up to the case of involutions of $E_6$ which requires a bit of theory of roots (this part is best looked up in Loos' book). |
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You can start with the chapter in the second volume of Kobayashi-Nomizu; both volumes are written in a reader friendly manner, and normally proofs are well detailed with precise references to the previous used results. Helgason is quite tough I think. Neither of these two are geometric enough in my opinion. The best for that are Loos's books (Symmetric space 1 and 2); these are my favourite ! |
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As a very basic reference, you might enjoy the easy-to-read Arvanitoyeorgos' book (see Chapter 6 for symmetric spaces): An Introduction to Lie Groups and the Geometry of Homogeneous Spaces (Student Mathematical Library, V. 22). Another good introductory text is the Chapter XI of Kobayashi & Nomizu, Foundations of Differential Geometry, vol II. Of course more advanced references are the ones mentioned before: Helgason, Wolf, Loos etc. |
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"Metric rigidity theorems on Hermitian locally symmetric manifolds" by Ngaiming Mok is good, but may not be basic. |
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