On Compact Hessian Manifolds

Abstract

We study compact Hessian manifolds. We prove vanishing of their Euler characteristics. For compact hyperbolic affine manifolds, we deduce that topologically they are periodic mapping tori, and that they admit no Riemannian metric of negative sectional curvature. We then establish a sharp integral inequality characterizing Cheng--Yau metrics. We also construct counterexamples to an incorrect but widely cited assertion of Shima--Yagi.

arXiv comment18 pages, 0 figure