Riemann curvature tensor | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-20

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Riemann_curvature_tensor


00:05:12 1 Geometric meaning
00:05:22 1.1 Informally
00:08:08 1.2 Formally
00:12:12 2 Coordinate expression
00:13:16 3 Symmetries and identities
00:23:07 4 Ricci curvature
00:30:59 5 Special cases
00:32:05 6 See also
00:36:10 7 Notes



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SUMMARY
=======
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.  It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space.  The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation.  The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
The curvature tensor is given in terms of the Levi-Civita connection 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
   by the following formula:

  
    
      
        R
        (
        u
        ,
        v
        )
        w
        =
        
          ∇
          
            u
          
        
        
          ∇
          
            v
          
        
        w
        −
        
          ∇
          
            v
          
        
        
          ∇
          
            u
          
        
        w
        −
        
          ∇
          
            [
            u
            ,
            v
            ]
          
        
        w
      
    
    {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w}
  where [u,v] is the Lie bracket of vector fields.  For each pair of tangent vectors u, v,  R(u,v) is a linear transformation of the tangent space of the manifold.  It is linear in u and v, and so defines a tensor.  Occasionally, the curvature tensor is defined with the opposite sign.
If 
  
    
      
        u
        =
        ∂
        
          /
        
        ∂
        
          x
          
            i
          
        
      
    
    {\displaystyle u=\partial /\partial x^{i}}
   and 
  
    
      
        v
        =
        ∂
        
          /
        
        ∂
        
          x
          
            j
          
        
      
    
    {\displaystyle v=\partial /\partial x^{j}}
   are coordinate vector fields then 
  
    
      
        [
        u
        ,
        v
        ]
        =
        0
      
    
    {\displaystyle [u,v]=0}
   and therefore the formula simplifies to  

  
    
      
        R
        (
        u
        ,
        v
        )
        w
        =
        
          ∇
          
            u
          
        
        
          ∇
          
            v
          
        
        w
        −
        
          ∇
          
            v
          
        
        
          ∇
          
            u
          
        
        w
        .
      
    
    {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w.}
  The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space).  The linear transformation 
  
    
      
        w
        ↦
        R
        (
        u
        ,
   ...