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A About diagonalisation of shape operators

  1. Apr 22, 2016 #1
    Dear fiends,
    I have a question if it possible,

    Given a submanifold Mⁿ of codimensioon p in a riemannian manifold N^{n+p}. Let (e₁,...,e_{p}) an orthonormal basis of the notmal vector space of Mⁿ in M^{n+p}.
    For every e_{i} we can define the shape opeator A_{e_{i}} of the second fundamental form correponding to the normal vector e_{i}.
    We know that each sahpe operator A_{e_{i}} is diagonalized.
    My question is : Is there a basis that diagonalized all the shap operators A_{e_{i}} simultaniously (at the same time)? And if the response is yes, is there any method, theorem or idea to find this basis?

    Thank you
    Last edited by a moderator: Apr 27, 2016
  2. jcsd
  3. Apr 27, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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