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I want to study basic properties of symmetric spaces.

What is a basic textbook?

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It would help people to give better answers if you said a bit about your background or existing knowledge, e.g. Lie group theory, Riemannian geometry, PDEs, or whatever –  Yemon Choi Sep 28 '11 at 3:21
Someone is going to mention Helgason's... –  Mariano Suárez-Alvarez Sep 28 '11 at 3:28
...and to feel guilty immediately afterwards –  Piero D'Ancona Sep 28 '11 at 6:27
What's wrong with Helgason? :) –  José Figueroa-O'Farrill Sep 28 '11 at 8:05
Helgason is awesome, but not really "basic"... –  Suvrit Sep 28 '11 at 8:32

5 Answers 5

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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