# Symmetries of non-Riemannian curvature tensor

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?

• I think you should look at mathoverflow.net/questions/69374/… – Ben McKay Jul 31 '15 at 20:20
• Thank you @BenMcKay, it is an interesting post. However, the trace for constructing the Ricci tensor is different than the one I mention. Additionally, the trace in the last two indices vanishes for Riemannian manifolds. – Dox Aug 1 '15 at 2:15
• Ben, this doesn't look like Ricci to me. The result is a skew-symmetric tensor and vanishes if the connection is the Levi-Civita connection. – Deane Yang Aug 1 '15 at 2:19
• Do you have any thoughts about what kind of condition you're looking for? – Deane Yang Aug 1 '15 at 4:52

Part of the confusion is that your positional notational convention is not the standard one. In most books, what you are writing as $R_{ab}{^c}_d$ would be written as $R{^c}_{dab}=-R{^c}_{dba}$ (though the letters used would probably be different).
That said, the condition $R{^c}_{cab}=0$ is the condition that the connection (locally) admit a parallel volume form. In other words, $R{^c}_{cab}$ represents a $2$-form that is the curvature of the induced connection on the line bundle $\Lambda^n(T^*M)$ (where $n$ is the dimension of the manifold). In particular, it is the exterior derivative of the $1$-form that you are writing as $\Gamma{_b}^{c}{_c}$. This $1$-form need not be zero (which would be $\Gamma{_b}^{c}{_c}=0$) nor need its coefficients be constant (which would be $\partial_a\Gamma{_b}^{c}{_c}=0$). It just needs to be closed in order to have your condition on the curvature.