The topology of 3-dimensional Hessian manifolds

TL;DR

This paper classifies the global topology of 3-dimensional Hessian manifolds, showing they are either Hantzsche-Wendt or Kähler mapping tori, and provides a complete topological classification of their structure.

Contribution

It offers a comprehensive topological classification of compact, orientable 3D Hessian manifolds, linking them to known geometric structures and analyzing their Betti numbers.

Findings

Compact 3D Hessian manifolds are either Hantzsche-Wendt or Kähler mapping tori.

The product of two Koszul type Hessian manifolds admits a Kähler structure.

All such manifolds are Seifert manifolds with specific orbifold Euler characteristics.

Abstract

We investigate the global topology of 3-dimensional Hessian manifolds. We prove that any compact, orientable 3-dimensional Hessian manifold is either a Hantzsche-Wendt manifold or admits the structure of a K\"ahler mapping torus. We analyze the parity of Betti numbers for compact, orientable 3-dimensional Hessian manifolds, with special focus on those of Koszul type (hyperbolic manifolds). Moreover, we show that the product of two compact, orientable, 3-dimensional Hessian manifolds of Koszul type naturally carries a K\"ahler structure. Finally, we establish that every compact, orientable, 3-dimensional Hessian manifold is a Seifert manifold with trivial Euler number, whose underlying orbifold has either vanishing or negative Euler characteristic, thus providing a complete topological classification.