Intrinsic characterization of Ricci tensor Picture below is from do Carmo's Riemannian Geometry.
First, what is intrinsic characterization ?   I  google it and fail to find key.




The first definition of $\operatorname{Ric}_p(x)$ involves choosing an orthonormal basis $z_1, \dots, z_{n-1}$ for $\{ x \}^{\perp} \subseteq T_pM$ and then plugging into some formula. Since there are infinitely many such orthonormal bases, the definition might a priori depend on the specific choice of basis.
The second definition of $\operatorname{Ric}_p(x)$ defines it as $\frac{Q_p(x,x)}{n-1}$ where $Q_p$ is a bilinear form on $T_pM$ defined "naturally" in terms of $R_p$ without any choice of a basis. This is what Do Carmo means by "intrinsically defined". The fact that both definitions are equivalent shows that the first definition actually does not depend on the basis used.
