Riemann symmetric space | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-02-18

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Symmetric_space


00:03:04 1 Definition using geodesic symmetries
00:04:54 1.1 Basic properties
00:06:06 1.2 Examples
00:07:00 2 General definition
00:12:27 3 Riemannian symmetric spaces satisfy  the Lie-theoretic characterization
00:16:45 4 Classification of Riemannian symmetric spaces
00:17:56 4.1 Classification scheme
00:22:56 4.2 Classification result
00:23:36 4.3 As Grassmannians
00:24:27 5 Symmetric spaces in general
00:25:44 5.1 Classification results
00:26:28 5.2 Tables
00:32:25 6 Weakly symmetric Riemannian spaces
00:33:08 7 Applications and special cases
00:33:52 7.1 Symmetric spaces and holonomy
00:35:17 7.2 Hermitian symmetric spaces
00:35:28 7.3 Quaternion-Kähler symmetric spaces
00:35:57 7.4 Bott periodicity theorem
00:37:31 8 See also
00:38:35 9 References
00:38:50 4 or q
00:39:02 1 or q
00:39:19 Bott periodicity theorem
00:39:41 See also



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SUMMARY
=======
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be made more precise, in either the language of Riemannian geometry or of Lie theory. The Riemannian definition is more geometric, and plays a deep role in the theory of holonomy. 
The Lie-theoretic definition is more algebraic.
In Riemannian geometry, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
   of M at p by minus the identity. 
Every symmetric space is complete, and has a finite cover which is a simply connected symmetric space; thus these two characterizations in fact coincide up to finite covers. Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.
From the point of view of Lie theory, a symmetric space is the quotient G/H of Lie group G by a Lie subgroup H, where the Lie algebra 
  
    
      
        
          
            h
          
        
      
    
    {\displaystyle {\mathfrak {h}}}
   of H is also required to be the +1-eigenspace of an involution of the Lie algebra 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
   of G. 
As stated, this characterization includes pseudo-Riemannian spaces as well as a Riemannian ones; extra algebraic conditions are needed to restrict to the Riemannian case.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first classified by Élie Cartan. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.