The expression [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] transforms as a symmetric (0, 2) tensor under projective coordinate changes of a domain in Rn 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 Kählergeometry. Let M be a compact locally projectively flat manifold. We prove that if u is a negative section of the dual of the tautological bundle such that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] is a Riemannian metric, then M is projectively equivalent to a quotient of a bounded convex domain in Rn. 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.


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pp. 595-609
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