Chern–Gauss–Bonnet theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 18/02/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Generalized_Gauss%E2%80%93Bonnet_theorem


00:02:18 1 Example: four dimensional manifolds
00:03:51 2 Further generalizations
00:05:44 3 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8342942844135814
Voice name: en-AU-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


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.