Hessian manifolds of constant Hessian sectional curvature
Let \(M\) be a flat affine manifold with a flat affine connection \(D\). If \(M\) admits a Riemannian metric \(g\) such that \(g\) is locally expressed by \(g= D^2 u\), then \((M, D, g)\) is said to be a Hessian manifold. We define Hessian sectional curvatures (these correspond to holomorphic sectional curvatures for Kählerian manifolds) and construct Hessian manifolds of constant Hessian sectional curvatures. We show that the family of normal distributions and the family of multinomial distributions are Hessian manifolds of constant Hessian sectional curvatures. We define affine Chern classes for flat affine manifolds (analogous to Chern classes for complex manifolds), and using affine Chern classes we give a characterization of constant Hessian sectional curvatures being similar to a result of Chen and Ogiue for complex space forms.
scientific article; zbMATH DE number 894107