Hessian Manifolds and Convexity

Abstract

Let M be a flat affine manifold, that is, M admits open charts (Ui, x i 1 ,..., x i n such that M =U Ui and whose coordinate changes are all affine functions. Such local coordinate systems { i n ,...,x i n } will be called affine local coordinate systems. Throughout this note the local expressions for geometric concepts on M will be given in terms of affine local coordinate systems. A Riemannian metric g on M is said to be Hessian if for each point p∈M there exists a C∞-function (⌽ defined on a neighborhood of p such that gij=∂2o/∂xi ∂xj. Such a function ⌽ is called a primitive of g on a neighborhood of p. Using the flat affine structure we define the exterior differentiation dl for tensor bundle valued forms on M. Let g be the cotangent bundle valued 1-form on M corresponding to a Riemannian metric g on M. Then we know that g is Hessian if and only if g0 is dl-closed. A flat affine manifold provided with a Hessian metric is called a Hessian manifold[4], [5]. Koszul dealt with the case where g0 is dl-exact [1], [2], [3].