  The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2)
tensor under projective coordinate transformations, so long as u transforms as
a section of a certain line bundle. On a locally projectively flat manifold M,
the section u can be regarded as a metric potential analogous to the local
potential in Kahler geometry. If M is compact and u is a negative section of
the dual of the tautological bundle whose Hessian is positive definite, then M
is projectively equivalent to a quotient of a bounded convex domain in R^n. The
same is true if M has a boundary on which u=0. This theorem is analogous to a
result of Schoen and Yau in locally conformally flat geometry. The proof uses
affine differential geometry techniques developed by Cheng and Yau.
