On Topology of Compact Hessian Manifolds

For a differentiable manifold $M$, a pair $(M,\mathrm{\nabla })$ is called an affine manifold if $\mathrm{\nabla }$ is a flat and torsion-free connection on the tangent bundle $TM\to M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on $(M,\mathrm{\nabla })$ if the $(0,3)$-tensor $\mathrm{\nabla }g$ is totally symmetric. We investigate general topological properties of compact Hessian manifolds, in particular confirm a long-standing conjecture of S.S.Chern in this special case, and then apply these constraints to low dimensions, providing a thorough topological classification of complete Hessian surfaces and closed orientable Hessian manifolds of dimension less than $5$.