Riemannian metrics on locally projectively flat manifolds

The expression -(1)/(u)u(ij) transforms as a symmetric (0, 2) tensor under projective coordinate changes of a domain in R-n so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold, the section u can be regarded as a metric potential analogous to the local potential in Kahler geometry. Let M be a compact locally projectively flat manifold. We prove that if it is a negative section of the dual of the tautological bundle such that -(1)/(u)u(ij) is a Riemannian metric, then M is projectively equivalent to a quotient of a bounded convex domain in R-n. The same is true for such manifolds M with boundary if u = 0 on the boundary. This theorem is an analog of a result of Schoen and Yau in locally conformally flat geometry. The proof involves affine differential geometry techniques developed by Cheng and Yau.