The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.

By construction it is antisymmetric in the first two indices, since roughly speaking $[\nabla_a , \nabla_b] \simeq R_{ab}{}^\bullet{}_\bullet$. I'm assuming the connection has vanishing torsion.

Question

Under what conditions $R_{ab}{}^c{}_c$ vanishes for a general curvature?

For example:

• If $\Gamma_{b}{}^c{}_c = 0$ the trace of $R$ vanishes.
• If $\partial_a \Gamma_{b}{}^c{}_c = 0$ the trace of $R$ vanishes.

Is there something more general? and Could that be interpreted as a gauge fixing?