On Topology of Compact Hessian Manifolds

For a differentiable manifold $M$, a pair $(M, \nabla)$ is called an affine manifold if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on $(M, \nabla)$ if the $(0,3)$-tensor $\nabla g$ 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 $5$.