
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 6,704 Thanks: 669 
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: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond 
Moved to Topology.


Tags 
diagonalisation, operators, shape 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Operators on inner product spaces  Luiz  Linear Algebra  2  October 7th, 2015 05:08 PM 
Rational operators  jamesuminator  Number Theory  10  December 12th, 2010 06:41 PM 
Example of operators  karkusha  Real Analysis  1  November 4th, 2010 12:23 AM 
New Operators  brangelito  Number Theory  7  April 26th, 2010 02:08 PM 
Compact Operators  Nusc  Real Analysis  1  April 11th, 2009 06:14 PM 