Generalized Gauss-Bonnet theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Generalized_Gauss%E2%80%93Bonnet_theorem


00:02:02 1 Example: four dimensional manifolds
00:03:25 2 Further generalizations
00:05:07 3 See also



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SUMMARY
=======
In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss and Pierre Ossian Bonnet) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of  the Gauss–Bonnet theorem (named after Carl Friedrich Gauss and Pierre Ossian Bonnet) to higher dimensions.
Let M be a compact orientable 2n-dimensional Riemannian manifold without boundary, and let 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
   be the curvature form of the Levi-Civita connection. This means that 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
   is an 
  
    
      
        
          
            s
          
        
        
          
            o
          
        
        (
        2
        n
        )
      
    
    {\displaystyle {\mathfrak {s}}{\mathfrak {o}}(2n)}
  -valued 2-form on M.  So 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
   can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring 
  
    
      
        
          ∧
          
            
              even
            
          
        
        
        
          T
          
            ∗
          
        
        M
      
    
    {\displaystyle \wedge ^{\hbox{even}}\,T^{*}M}
  . One may therefore take the Pfaffian 
  
    
      
        
          
            Pf
          
        
        (
        Ω
        )
      
    
    {\displaystyle {\mbox{Pf}}(\Omega )}
  , which turns out to be a 2n-form.
The generalized Gauss–Bonnet theorem states that

  
    
      
        
          ∫
          
            M
          
        
        
          
            Pf
          
        
        (
        Ω
        )
        =
        (
        2
        π
        
          )
          
            n
          
        
        χ
        (
        M
        )
         
      
    
    {\displaystyle \int _{M}{\mbox{Pf}}(\Omega )=(2\pi )^{n}\chi (M)\ }
  where 
  
    
      
        χ
        (
        M
        )
      
    
    {\displaystyle \chi (M)}
   denotes the Euler characteristic of M.