A Riemannian metric on a flat manifold is called a "Hessian metric" if it is locally expressed by the Hessian of functions with respect to the affine coordinate systems. A pair of a flat structure and a Hessian metric is called a "Hessian structure," and a manifold with a Hessian structure is called a "Hessian manifold." This new field of research, which is closely allied to Kählerian geometry, is also related to affine differential geometry, homogeneous spaces and cohomology. Shima (mathematics, Yamaguchi U.) focuses on theory as he explains affine spaces and connections, Hessian structures, including their relations with Kählerian and Codazzi structures, curvatures for Hessian structures, regular convex cones, Hessian structures and affine differential geometry, Hessian structures and information geometry, the cohomology of flat manifolds, compact Hessian manifolds, symmetric spaces with invariant Hessian structures, homogeneous spaces with invariant Hessian structures, and homogeneous spaces with invariant projectively flat connections. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)