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Diffeomorphism between the Klein bottle and the connected sum of projective spaces

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2

$begingroup$

I know that the Klein bottle is a $C^{infty}$ manifold (obtained as the quotient space of the action $(x,y) longrightarrow (x+1/2,-y)$ on the torus).

I have to find the diffeomorphim between the so obained Klein bottle and the connected sum of two projective spaces. I know it can be seen "cutting and pasting" the square which represents the Klein bottle $aba^{-1}b$, however I would like to find an esplicit diffeomormism, is that possible?

differential-geometry

share|cite|improve this question

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  Mar 18 at 20:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Capitalize the initial letter of a name like Klein (klein is small :) anglo-german pun)
  $endgroup$
  – Jean Marie
  Mar 18 at 20:30
  

  
  
  
  

add a comment | 

2

$begingroup$

I know that the Klein bottle is a $C^{infty}$ manifold (obtained as the quotient space of the action $(x,y) longrightarrow (x+1/2,-y)$ on the torus).

I have to find the diffeomorphim between the so obained Klein bottle and the connected sum of two projective spaces. I know it can be seen "cutting and pasting" the square which represents the Klein bottle $aba^{-1}b$, however I would like to find an esplicit diffeomormism, is that possible?

differential-geometry

share|cite|improve this question

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  Mar 18 at 20:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Capitalize the initial letter of a name like Klein (klein is small :) anglo-german pun)
  $endgroup$
  – Jean Marie
  Mar 18 at 20:30
  

  
  
  
  

add a comment | 

$begingroup$

I know that the Klein bottle is a $C^{infty}$ manifold (obtained as the quotient space of the action $(x,y) longrightarrow (x+1/2,-y)$ on the torus).

I have to find the diffeomorphim between the so obained Klein bottle and the connected sum of two projective spaces. I know it can be seen "cutting and pasting" the square which represents the Klein bottle $aba^{-1}b$, however I would like to find an esplicit diffeomormism, is that possible?

differential-geometry

share|cite|improve this question

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

$endgroup$

share|cite|improve this question

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

share|cite|improve this question

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

edited Mar 18 at 20:29

Jean Marie

31.2k42255

edited Mar 18 at 20:29

Jean Marie

31.2k42255

asked Mar 18 at 19:58

dennisdennis

111

asked Mar 18 at 19:58

dennisdennis

111