I am looking for the integrability condition of the following system of pde:

$$\partial_{[\nu}\Gamma^\kappa_{\mu]\lambda}+\Gamma^\kappa_{[\nu|\rho|}\Gamma^\rho_{\mu]\lambda}=\frac{1}{2}R_{\mu\nu\lambda}{}^{\kappa},\,\,\,\,\,\,\,\,\,(1)$$

given sufficiently smooth functions $R_{\mu\nu\lambda}{}^{\kappa}$ on some smooth manifold $M^n$, with $R_{(\mu\nu)\lambda}{}^{\kappa}=0$. This equation is the definition of the Riemann-Christoffel curvature of a *given* connection $\Gamma^\lambda_{\mu\nu}$. But I am interested in the reverse questions:

*Given functions $R_{\mu\nu\lambda}{}^{\kappa}$, with $R_{(\mu\nu)\lambda}{}^{\kappa}=0$, what are the conditions such that they form the compoments of the Riemann-Christoffel curvature of some connection?**If the given functions $R_{\mu\nu\lambda}{}^{\kappa}$ do satisfy those integrability conditions, then what will be general form of the connection coefficients $\Gamma^\lambda_{\mu\nu}$?*(When $R_{\mu\nu\lambda}{}^{\kappa}=0$, it is well-known that there exists an invertible matrix field $[A_{\mu\nu}]$ (with inverse $[A^{\mu\nu}]$) such that $\Gamma^\gamma_{\mu\nu}=A^{\gamma\lambda}\partial_{\nu} A_{\lambda\mu}$.)

In Schouten (Chapter III, equation 5.19), it is stated that the integrability condition of (1) are given by the Bianchi identities

$$\nabla_{[\omega}R_{\mu\nu]\lambda}{}^\kappa=2S_{[\omega\nu}{}^\sigma\, R_{\mu]\sigma\lambda}{}^\kappa,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$$ where $S_{\mu\nu}{}^\kappa$ is the torsion of the connection $\Gamma^\kappa_{\mu\nu}$.

It is clear that (2) involves covarient derivative wirh respect to the connection $\Gamma$ and also the torsion $S$ of $\Gamma$, but to start with no such $\Gamma$ is given. We are *looking for* such a $\Gamma$. I fail to understand in what sense, (2) is an integrability condition of (1). I'm also not sure, whether to determine a $\Gamma$ satisfying (1), some (skew-symmetric) functions $S_{\mu\nu}{}^\lambda$ should also be given *a priori* so that the desired connection has also these specified fuctions as its torsion.

**Notes:** 1. No metric structure is given on the manifold $M^n$.

- It would be great if it can be answered at least for $n=3$.

Thanks in advance.

anymatrix of one form $\Gamma$, not necessarily the connection form $\Gamma$ which we are seeking. Am I correct? $\endgroup$ – Ayan May 25 '15 at 6:44