Hilbert's theorem (differential geometry) | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Hilbert%27s_theorem_(differential_geometry)


00:00:52 1 History
00:01:35 2 Proof
00:21:55 3 See also



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SUMMARY
=======
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface 
  
    
      
        S
      
    
    {\displaystyle S}
   of constant negative gaussian curvature 
  
    
      
        K
      
    
    {\displaystyle K}
   immersed in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  . This theorem answers the question for the negative case of which surfaces in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
   can be obtained by isometrically immersing complete manifolds with constant curvature.