Riemannian Metrics on Locally Projectively Flat Manifolds
Abstract
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2001
 arXiv:
 arXiv:math/0108218
 Bibcode:
 2001math......8218L
 Keywords:

 Differential Geometry;
 57N16 (Primary);
 53A15 (Secondary)
 EPrint:
 16 pages, to be published in American Journal of Mathematics