On Hessian Structures on an Affine Manifold

Abstract

On a smooth manifold, an affine connection whose torsion and curvature vanish identically is called an affine structure. A smooth manifold provided with an affine structure is called an affine manifold. Let M be an affine manifold with an affine structure D. The co-variant differentiation by D will be also denoted by D. A Riemannian metric h on M is called a hessian metric if for each point x∈M there exist a neighborhood U of x and a smooth function ⌽ on U such that g = D2⌽ on U [5]. In this note we shall give an example of an affine manifold which does not admit any hessian metric and then determine the structure of A-Lie algebras which admit hessian metrics. For these purposes, we shall also establish a vanishing theorem of a certain cohomology group. The author would like to thank Professor H. Shima who introduced him to the problem discussed here.