### On Locally Symmetrie Homogeneous Domains of Completely Reducible Linear Lie Groups.

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A manifold is said to be Hessian if it admits a flat affine connection $D$ and a Riemannian metric $g$ such that $g={D}^{2}u$ where $u$ is a local function. We study cohomology for Hessian manifolds, and prove a duality theorem and vanishing theorems.

A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form ${g}_{ij}=\frac{{\partial}^{2}\Phi}{\partial {x}^{i}\partial {x}^{j}}$ where $\Phi $ is a ${C}^{\infty}$-function and $\{{x}^{1},...,{x}^{n}\}$ is an affine local coordinate system. Let $M$ be a Hessian manifold. We show that if $M$ is homogeneous, the universal covering manifold of $M$ is a convex domain in ${\mathbf{R}}^{n}$ and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of ${\mathbf{R}}^{n}$.

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