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I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained from my library the "Lectures on Closed Geodesics" by Klingenberg, but found it to be rather dry and difficult to read. Do you have any other suggestions?

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  • $\begingroup$ What would you like to learn about free loop spaces, specifically? $\endgroup$ – Ryan Budney Mar 17 '15 at 3:05
  • $\begingroup$ I'd like a thorough text that contains the foundations of the subject, but more specifically the fact that it satisfies condition (C) of Palais and Smale. If possible, I would like a text that introduces Morse Theory on this space, since it is what the paper is about. $\endgroup$ – Aloizio Macedo Mar 17 '15 at 3:09
  • $\begingroup$ Have you looked at Milnor's Morse Theory book? $\endgroup$ – Ryan Budney Mar 17 '15 at 3:10
  • $\begingroup$ Thank you, I did not know that this subject was also in this book... but does it deal with the free loop space, or only "similar spaces"? I am not with the book right now, so I can't check... Furthermore, do you think I should try harder with Klingenberg, or is it for a second reading? $\endgroup$ – Aloizio Macedo Mar 17 '15 at 3:19
  • $\begingroup$ Klingenberg's Riemannian Geometry should be easier and has also some material about the free loop space and Hilbert manifolds. $\endgroup$ – Lennart Meier Mar 17 '15 at 3:24
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This survey paper on the Morse theory and closed geodesics: http://arxiv.org/abs/1406.3107 (Morse theory, closed geodesics, and the homology of free loop spaces, by Alexandru Oancea) provides a lot of relevant references. In particular, it says that "a beautiful reference concerning closed geodesics is the book by Besse": http://link.springer.com/book/10.1007/978-3-642-61876-5 (Manifolds all of whose Geodesics are Closed).

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