Why the Ricci tensor is defined the contraction of first and third indices of Riemann tensor? I guess it is more natural to define it as to contract the first and the second indices?
Since from Wiki :
$$R_{\sigma \mu \nu}^\rho = dx^\rho(R(\partial_\mu,\partial_\nu)\partial_\sigma)$$
To do a contraction, I guess it should be done on $\rho $ and $\sigma$ so that
$$R_{\mu \nu} = R_{\sigma \mu \nu}^ \sigma$$
but why the definition is
$$R_{\mu \nu} = R_{\mu \sigma \nu}^ \sigma$$
Best Answer
The symmetries of the Riemann tensor imply that $R^\sigma{}_{\sigma \mu \nu} = 0$. In fact the Ricci curvature is the only non-zero contraction of the Riemann tensor.