Ricci Curvature and Betti Numbers of Hessian Manifolds

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.