Abstract
A symmetric space is a connected Riemannian manifold M where for each point p ∈ M there is an isometry σ p of M such that σ p (p) = p and the differential of σ p at p is multiplication by —1. Symmetric spaces were introduced by Elie Cartan in 1926 [Car26] and are generally regarded as being among the most fundamental and beautiful objects in mathematics; they play a fundamental role in the theory of semi-simple Lie groups and enjoy many remarkable properties. A comprehensive treatment of symmetric spaces is beyond the scope of this book, but we feel that there is considerable benefit in describing certain key examples from scratch (without assuming any background in differential geometry or the theory of Lie groups), in keeping with the spirit of the book. Simple examples of symmetric spaces include the model spaces M n κ that we studied in Part I.
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© 1999 Springer-Verlag Berlin Heidelberg
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Bridson, M.R., Haefliger, A. (1999). Symmetric Spaces. In: Metric Spaces of Non-Positive Curvature. Grundlehren der mathematischen Wissenschaften, vol 319. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12494-9_18
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DOI: https://doi.org/10.1007/978-3-662-12494-9_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08399-0
Online ISBN: 978-3-662-12494-9
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