Riemannian Metrics on Locally Projectively Flat Manifolds

2001/08/30 by J. C. Loftin, Loftin, J. C.
Mathematics · #53A15 (Secondary) #57N16 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A15 #msc:57N16

paper · pdf · doi:10.48550/arxiv.math/0108218

16 pages, to be published in American Journal of Mathematics

arxiv created 2001/08/30 · arxiv updated 2009/11/30

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 Rⁿ. 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.

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