About diagonalisation of shape operators

[aboutammam]

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

 

[jvicens]

for your question. Yes, it is possible to find a basis that diagonalizes all the shape operators simultaneously. This is known as the principal directions or principal basis. The principal directions are the eigenvectors of the shape operators, and the corresponding eigenvalues are the principal curvatures. This means that the shape operators can be written as diagonal matrices with the principal curvatures along the diagonal.

To find the principal directions, you can use the Jacobi method or the Jacobi-Davidson method. These are iterative methods that can efficiently compute the principal directions and principal curvatures. Another method is the spectral decomposition method, which involves finding the eigenvalues and eigenvectors of the shape operators.

In terms of theorems, there is the Spectral Theorem for symmetric matrices, which states that a symmetric matrix can be diagonalized by an orthogonal matrix. Since the shape operators are symmetric, this theorem can be applied to find the principal directions.

I hope this helps answer your question. Let me know if you need any further clarification or assistance. Good luck with your research!