The geometry of Hessian structures is a fascinating emerging field of research. It is in particular a very close relative of Kählerian geometry, and connected with many important pure mathematical branches such as affine differential geometry, homogeneous spaces and cohomology. The theory also finds deep relation to information geometry in applied mathematics. This systematic introduction to the subject first develops the fundamentals of Hessian structures on the basis of a certain pair of a flat connection and a Riemannian metric, and then describes these related fields as applications of the theory.
Sample Chapter(s)
Chapter 1: Affine spaces and connections (150 KB)
Contents:
- Affine Spaces and Connections
- Hessian Structures
- Curvatures for Hessian Structures
- Regular Convex Cones
- Hessian Structures and Affine Differential Geometry
- Hessian Structures and Information Geometry
- Cohomology on Flat Manifolds
- Compact Hessian Manifolds
- Symmetric Spaces with Invariant Hessian Structures
- Homogeneous Spaces with Invariant Hessian Structures
- Homogeneous Spaces with Invariant Projectively Flat Connections
Readership: Mathematicians and mathematics graduate students.
