About diagonalisation of shape operators

  • #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
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