Homogeneous hessian manifolds

Abstract

A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form g i j = ∂ 2 Φ ∂ x i ∂ x j where Φ is a C ∞ -function and { x 1 , ... , x n } is an affine local coordinate system. Let M be a Hessian manifold. We show that if M is homogeneous, the universal covering manifold of M is a convex domain in R n and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of R n .