Index gymnastics with the Riemann tensor

Question

We would like to obtain the Ricci tensor from the Riemann tensor. In most books are contracted the first index with the third one, the second index with the fourth one. Following the same convention, I have obtained these identities. $$g^{ia}g^{lm}{R^k}_{lam}=g^{ia}{R^k}_a=R^{ik}$$ $$g^{im}g^{la}{R^k}_{lam}=-g^{im}g^{la}{R^k}_{lma}=-g^{im}{R^k}_m=-R^{ik}$$ Are they correct?

Answer 1

I am not sure why you want the Ricci tensor to have upper indices. If you don't insist on that, but use the more natural version with two lower indices,things become easier: You have to contact the (unique) upper index into one of the two-form indices of the curvature. (Here I am using the interpretation that the Riemann tensor is a two-form with values in endomorphisms of the tangent bundle, which evan available for linear connections without a Riemannian metric.) This fixes Ricci up to sign. The sign is indeed a subtle issue and needs writing out the conventions you use for the Riemann-tensor.