Hessian Manifolds and Convexity

Abstract

Let M be a flat affine manifold, that is, M admits open charts (U_i, x ¹_i ,..., x ⁿ_i such that M =U U_i and whose coordinate changes are all affine functions. Such local coordinate systems { ⁿ_i ,...,x ⁿ_i } 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 g_ij=∂²ø/∂xⁱ ∂x^j. 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].