I am searching for a tensor in 4-dimensional space-time with two indices that satisfy:
\begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0, \\ M^{\mu \nu } + M^{\nu\mu}&=&0, \nonumber \\ M_{;\varepsilon }^{\mu \nu }+M_{;\nu }^{\varepsilon \mu }+M_{;\mu }^{\nu \varepsilon } &=&0. \nonumber \end{eqnarray}
An obvious choice would be the electromagnetic field strength, but I am investigating if it is possible to build a tensor with such properties that only depends on geometrical properties of the space-time manifold (metric tensor, connection). For instance, the Riemann tensor satisfies two of these properties for fixed α, β:
\begin{eqnarray} R_{;\varepsilon }^{\alpha \beta \mu \nu }+R_{;\nu }^{\alpha \beta \varepsilon \mu }+R_{;\mu }^{\alpha \beta \nu \varepsilon } &=&0\text{ } \\ R^{\alpha \beta \mu \nu } &=&-R^{\alpha \beta \mu \nu } \nonumber \end{eqnarray}
But I found its divergence is zero only in very special cases, such as for maximally symmetric spaces. Is there anyway to build a tensor with these properties always, or at least in not so special cases?
If you lower the indices and consider $$M_{\mu \nu} = g_{\gamma \mu}g_{\delta \nu}M^{\gamma \delta},$$ then $M_{\mu \nu}$ is a two form (by equation two), is co-closed (first equation) and closed (third equation). Thus you are looking for a harmonic two form. (I am treating these as Riemannian manifolds, so I hope I did not misunderstand the concepts).