The Geometry of Hessian Structures

Hirohiko Shima9812700315, 9789812700315, 9789812707536

The geometry of Hessian structures is a fascinating emerging field of research. It is in particular a very close relative of Kahlerian 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.

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Table of contents :
Contents……Page 14
Preface……Page 8
Introduction……Page 10
1.1 Affine spaces……Page 16
1.2 Connections……Page 19
1.3 Vector bundles……Page 24
2.1 Hessian structures……Page 28
2.2 Hessian structures and Kahlerian structures……Page 33
2.3 Dual Hessian structures……Page 37
2.4 Divergences for Hessian structures……Page 44
2.5 Codazzi structures……Page 47
3.1 Hessian curvature tensors and Koszul forms……Page 52
3.2 Hessian sectional curvature……Page 58
4.1 Regular convex cones……Page 68
4.2 Homogeneous self-dual cones……Page 78
5.1 Affine hypersurfaces……Page 92
5.2 Level surfaces of potential functions……Page 97
5.3 Laplacians of gradient mappings……Page 108
6.1 Dual connections on smooth families of probability distributions……Page 118
6.2 Hessian structures induced by normal distributions……Page 125
7.1 (p, q)-forms on at manifolds……Page 130
7.2 Laplacians on at manifolds……Page 136
7.3 Koszul’s vanishing theorem……Page 139
7.4 Laplacians on Hessian manifolds……Page 144
7.5 Laplacian L……Page 153
7.6 Affine Chern classes of at manifolds……Page 156
8.1 Affine developments and exponential mappings for at manifolds……Page 164
8.2 Convexity of Hessian manifolds……Page 167
8.3 Koszul forms on Hessian manifolds……Page 175
9.1 Invariant at connections and affine representations……Page 180
9.2 Invariant Hessian structures and affine representations……Page 185
9.3 Symmetric spaces with invariant Hessian structures……Page 189
10.1 Simply transitive triangular groups……Page 198
10.2 Homogeneous regular convex domains and clans……Page 202
10.3 Principal decompositions of clans and real Siegel domains……Page 208
10.4 Homogeneous Hessian domains and normal Hessian algebras……Page 223
11.1 Invariant projectively at connections……Page 230
11.2 Symmetric spaces with invariant projectively at connections……Page 235
11.3 Invariant Codazzi structures of constant curvature……Page 243
Bibliography……Page 252
Index……Page 258