in this paper, we obtain two intrinsic integral inequalities of hessian manifolds.

We show that Hessian manifolds of dimensions 4 and above must have vanishing Pontryagin forms. This gives a topological obstruction to the existence of Hessian metrics. We find an additional explicit curvature identity for Hessian 4-manifolds. By contrast, we show that all analytic Riemannian 2-manifolds are Hessian.

A selfsimilar manifold is a Riemannian (M,g) endowed with homothetic vector field ξ. We characterize global manifolds and describe the structure of local manifolds. prove that any potential conical or Euclidean space. radiant Hessian (M,∇,g,ξ) such ∇ξ=λId, where λ constant. locally isomorphic to product

In 1985, Amari [1] introduced an interesting manifold, i.e., statistical manifold in the context of information geometry. The geometry of such manifolds includes the notion of dual connections, called conjugate connections in affine geometry, it is closely related to affine geometry. A statistical structure is a generalization of a Hessian one, it connects Hessian geometry. In the present paper...