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In Exercise 9, Chaper 4, do Carmo gives a hint to solving a problem. He says to consider an orthonormal basis $e_1, \ldots, e_n$ in $T_pM$ such that if $x = \sum_{i=1}^n x_i e_i$,

$$\text{Ric}_p(x) = \sum \lambda_i x_i^2,$$

$\lambda_i$ real. My question is, why must such an orthonormal basis exist?

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This can be done point by point and is a purely linear algebraic fact.

At a point, we can view Ric(U,V) as a symmetric bilinear form on the vector space $T_pM$. But symmetric bilinear forms are known to be orthogonally diagonalizable.

A proof can be found here, theorem 1.4. In the proof, the author Joel Kamnitz doesn't explicitly show the basis can be chosen to be orthonormal, but his proof can be trivially modified to show it. Specifically, change the inductive hypothesis from "there is a basis..." to "there is an orthonormal basis...".

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