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: 


1. *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?*
2. *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 $M^n$.


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


Thanks in advance.



Does the Riemann-Christoffel curvature determine the connection?
Part of the difficulty in providing an answer to your question is the fact that the expression "the integrability condition" is a somewhat vague notion, and it's used in slightly different senses in different contexts. 


The usual, somewhat imprecise, sense is that, for a given system of PDE, its 'integrability condition' is the set of necessary and sufficient conditions that it have local solutions. Slightly more precise usage would be, given a PDE system that depends on some data (in your case, the equations (1), where the data is the curvature $R$), to find the necessary and sufficient conditions *on the data* that ensure that the PDE system have local solutions.


In this sense, the Bianchi identities (your equation (2)) are usually not *the* integrability condition for the system (1) but are only *part* of the integrability condition, and they have to be interpreted appropriately. The point is that, when $R$ is specified, equation (2) is an *algebraic* equation $\Gamma$ must satisfy if it is to satisfy the first order PDE system (1). [It may not be immediately apparent that (2) is an algebraic equation on $\Gamma$, but if you look at the definition of $\nabla$, you'll see that the left hand side of (2) only involves the unknown coefficients of $\Gamma$ linearly with no differentiation of them, and the right hand side involves the unknown $S$, which is computed linearly from the unknown $\Gamma$.] Thus, a necessary condition *on the data $R$* that (1) have a solution is that there be at least one solution $\overline\Gamma$ to the inhomogeneous linear algebraic system (2) (whose coefficients come from $R$ and its first derivatives). Generally, this necessary condition is *not* sufficient, though, as examples in dimensions $4$ and above show.


However, what Deane Yang's reference eudml.org/doc/74779 (DeTurck & Talvacchia, 1987) shows is that, when (i) $n=3$, (ii) $R$ is sufficiently generic (in an appropriate sense) while satisfying the integrability condition (2), *and* (iii) $R$ is real-analytic, then the system (1) is locally solvable. The system of PDE for $\Gamma$ that has to be solved, though, is highly nonlinear and, in a sense, overdetermined, so that the Cartan-Kähler Theorem or one of its modern versions has to be applied. There is no uniqueness and no way explicitly to solve the equations for $\Gamma$ for a given generic $R$ that satisfies the integrability condition. In that sense, your second 'reverse question' has no good answer.


Finally, let me remark that one *could* conceivably want to answer the more restrictive problem of specifying *both* the curvature $R$ and torsion $S$ of a connection $\Gamma$ on a manifold $M$. This problem is overdetermined even when $n=3$ and, generally, has no solution. This problem also has an additional set of integrability conditions of the form $\nabla S = F(R,S)$ (for an explicit expression $F(R,S)$ that one can write down easily) and which are also inhomogeneous linear algebraic in $\Gamma$, in addition to the Bianchi conditions $\nabla R = G(R,S)$ that you already have written down as (2). In general, these combined integrability conditions are *not* sufficient for the generic pair $(S,R)$ that satisfy them in order for a solution $\Gamma$ to exist.


