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%From: Hirohiko Shima <shima@po.cc.yamaguchi-u.ac.jp>
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%Dear Professor Hyuk Kim,

%Thank you very much for you to picking me up at Kimpo airport.
%I send a short abstract of my lectures.
%I prefere to give lectures more than five times.

%Looking forward to seeing you soon in Seoul,

%Hirohiko Shima


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\documentclass[12pt,a4paper]{article}

%\usepackage{amsmath,amssymb}

\pagestyle{empty}

\begin{document}

\begin{center}
{\bf Geometry of Hessian manifolds}
\end{center}

\noindent  A \underline{{\it 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
\[ g_{ij}=\frac{\partial^2\varphi}{\partial x^i\partial x^j}
\]
with respect to affine coordinate system for $D$.
A manifold with Hessian structure is said to be a \underline{{\it
Hessian manifold}}.
A Hessian structure on $M$ induces a K\"ahlerian structure on the
tangent bundle
$TM$ over $M$, so Geometry of Hessian manifolds is deeply connected with
K\"ahlerian 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,...).
\vspace{3mm}

\underline{{\bf 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
   

\end{document}
