Contents -- Preface -- Introduction -- 1. Affine spaces and connections -- 1.1 Affine spaces -- 1.2 Connections -- 1.3 Vector bundles -- 2. Hessian structures -- 2.1 Hessian structures -- 2.2 Hessian structures and Kahlerian structures -- 2.3 Dual Hessian structures -- 2.4 Divergences for Hessian structures -- 2.5 Codazzi structures -- 3. Curvatures for Hessian structures -- 3.1 Hessian curvature tensors and Koszul forms -- 3.2 Hessian sectional curvature -- 4. Regular convex cones -- 4.1 Regular convex cones -- 4.2 Homogeneous self-dual cones -- 5. Hessian structures and affine differential geometry -- 5.1 Affine hypersurfaces -- 5.2 Level surfaces of potential functions -- 5.3 Laplacians of gradient mappings -- 6. Hessian structures and information geometry -- 6.1 Dual connections on smooth families of probability distributions -- 6.2 Hessian structures induced by normal distributions -- 7. Cohomology on at manifolds -- 7.1 (p, q)-forms on at manifolds -- 7.2 Laplacians on at manifolds -- 7.3 Koszul's vanishing theorem -- 7.4 Laplacians on Hessian manifolds -- 7.5 Laplacian L -- 7.6 Affine Chern classes of at manifolds -- 8. Compact Hessian manifolds -- 8.1 Affine developments and exponential mappings for at manifolds -- 8.2 Convexity of Hessian manifolds -- 8.3 Koszul forms on Hessian manifolds -- 9. Symmetric spaces with invariant Hessian structures -- 9.1 Invariant at connections and affine representations -- 9.2 Invariant Hessian structures and affine representations -- 9.3 Symmetric spaces with invariant Hessian structures -- 10. Homogeneous spaces with invariant Hessian structures -- 10.1 Simply transitive triangular groups -- 10.2 Homogeneous regular convex domains and clans -- 10.3 Principal decompositions of clans and real Siegel domains -- 10.4 Homogeneous Hessian domains and normal Hessian algebras.

11. Homogeneous spaces with invariant projectively at connections -- 11.1 Invariant projectively at connections -- 11.2 Symmetric spaces with invariant projectively at connections -- 11.3 Invariant Codazzi structures of constant curvature -- Bibliography -- Index.