Ricci Curvature and Betti Numbers of Hessian Manifolds

TL;DR

This paper investigates the Ricci curvature of Hessian manifolds, revealing that non-negative Ricci curvature on a leaf implies flatness and constrains Betti numbers, highlighting a rigidity phenomenon in affine geometry.

Contribution

It establishes a rigidity result linking leafwise Ricci curvature to flatness and topological bounds in Hessian manifolds, and classifies three-dimensional cases.

Findings

Non-negative Ricci curvature on a leaf implies the Hessian metric is flat.

Bounds on the first Betti number depend on the manifold's dimension and leaf topology.

No leaf of a Koszul-type or radiant affine manifold can have non-negative Ricci curvature.

Abstract

We study Ricci curvature properties of Hessian metrics on the leaves of the codimension-one foliation $\mathcal{F}_\omega = \ker\,\omega$ generated by the first Koszul form $\omega$ of a closed oriented Hessian manifold. Our main result reveals a striking rigidity phenomenon: non-negative Ricci curvature on a single leaf of $\mathcal{F}_\omega$ compels the Hessian metric to be flat, yields sharp bounds on the first Betti number in terms of the dimension of the Hessian manifold and the topology of the leaves. This rigidity also shows that Koszul-type and radiant affine manifolds admit no leaf carrying non-negative Ricci curvature, reflecting a fundamental incompatibility between affine hyperbolicity and leafwise curvature positivity. In dimension three, we obtain a complete classification of the underlying manifold, extended to the non-orientable setting via the orientation double cover.