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April 22nd, 2016, 06:37 AM  #1 
Newbie Joined: Oct 2015 From: algeria Posts: 5 Thanks: 0 
Hello friends, I have a question. Given a submanifold Mⁿ of codimension p in a Riemannian manifold N^{n+p}, let (e₁,...,e_{p}) be an orthonormal basis of the normal vector space of Mⁿ in M^{n+p}. For every e_{i}, we can define the shape operator A_{e_{i}} of the second fundamental form corresponding to the normal vector e_{i}. We know that each shape operator A_{e_{i}} is diagonalized. My question is: Is there a basis that diagonalized all the shape operators A_{e_{i}} simultaneously (at the same time)? And if the response is yes, is there any method, theorem or idea to find this basis? If it possible. Thank you. Last edited by skipjack; April 22nd, 2016 at 08:05 PM. 
April 22nd, 2016, 01:14 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,806 Thanks: 716 
Wrong forum  this is not a high school algebra question!
Last edited by skipjack; April 22nd, 2016 at 08:05 PM. 
April 22nd, 2016, 06:29 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
Moved to Topology.


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diagonalisation, operators, shape 
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