On Topology of Compact Hessian Manifolds
Hanwen Liu
TL;DR
The paper develops a topological theory for compact Hessian manifolds, relating affine structures, Hessian metrics, and information-geometric objects. It proves that complete Hessian manifolds are aspherical with convex developable universal covers and shows that compact Hessian manifolds have infinite, torsion-free fundamental groups, yielding a global potential criterion when $\pi_1(M)$ is finite. A central corollary confirms Chern’s conjecture in this setting by showing the Euler characteristic of any compact Hessian manifold vanishes and derives a broad fibration/rigidity framework for hyperbolic affine pieces. In low dimensions, the authors give a complete topological classification: complete Hessian surfaces are topologically flat, closed orientable Hessian 3-manifolds are mapping tori of periodic surface automorphisms, and closed orientable Hessian 4-manifolds are Bieberbach manifolds or certain mapping tori, with explicit constructions validating the classifications. The work also establishes decomposition results via finite covers, linking hyperbolic-affine factors with flat tori and providing a robust global picture of Hessian geometry from a topological viewpoint.
Abstract
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 $5$.