Geometry of Hessian manifolds

A \underline{Hessian structure} on a manifold is a pair (D, g) of a flat connection D and a Riemannian metric g such that g is locally expressed by Hessian

g_{ij} = \frac{∂^{2}φ}{∂x^{i}∂x^{j}}

with respect to affine coordinate system for D. A manifold with Hessian structure is said to be a \underline{Hessian manifold}. A Hessian structure on M induces a Kählerian structure on the tangent bundle T M over M, so Geometry of Hessian manifolds is deeply connected with Kählerian Geometry. It is quite interesting that Hessian Geometry is also related with Information Geometry (that is, many important smooth families of probability distributions admit Hessian structures, e.g. normal distributions, binomial distributions,...).

\underline{Contents}

1. Hessian structures

2. Hessian curvature tensor and Koszul form

3. Regular convex cones

4. Hessian structures and Information Geometry

5. Homogeneous Hessian manifolds

6. Homogeneous projectively flat manifolds

7. Affine Chern classes