I'm looking for a curvature tensor "$C$" that characterizes the curvature like the Riemann tensor but adds up linearly regarding the metric tensor:
$C_{\mu\nu} (g_1+g_2) = C_{\mu\nu} (g_1)+C_{\mu\nu} (g_2)$
$C^{l}_{ijk}(a\cdot g)= a\cdot C^{l}_{ijk}(g)$
Is such a tensor possible to construct? How? (if not, why not?)
The Riemann tensor itself does not fulfill these requests as for each (positive) constant $a$, $R^{l}_{ijk}(ag)= R^{l}_{ijk}(g)$ and $R_{\mu\nu} (g_1+g_2) \neq R_{\mu\nu} (g_1)+R_{\mu\nu} (g_2)$ (For details see this)