Lie Groups pp 257-280 | Cite as

Symmetric Spaces

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


We have devoted some attention to an important class of homogeneous spaces of Lie groups, namely flag manifolds. Another important class is that of symmetric spaces. In differential geometry, a symmetric space is a Riemannian manifold in which around every point there is an isometry reversing the direction of every geodesic. Symmetric spaces generalize the non-Euclidean geometries of the sphere (compact with positive curvature) and the Poincaré upper half-plane (noncompact with negative curvature). Like these two examples, they tend to come in pairs, one compact and one noncompact. They were classified by E. Cartan.


Symmetric Space Boundary Component Jordan Algebra Maximal Compact Subgroup Linear Fractional Transformation 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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