Does the Riemann-Christoffel curvature determine the connection?

Question

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.

Answer 1

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.

Answer 2

It seems a partial answer (the solution is found with some additional restrictions) is given in the paper http://link.springer.com/article/10.1007%2FBF00759087 (Determination of the metric from the curvature, by Hernando Quevedo)

P.S. Of course Deane Yang's answer/comment above is more relevant for your question. If you not already found it, see also http://www.sciencedirect.com/science/article/pii/S0021782409001317 (The prescribed curvature problem in dimension four, by J.M. Masqué, L.M. Pozo Coronado and I. Sánchez Rodríguez).