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The topology of 3-dimensional Hessian manifolds

Emmanuel Gnandi

Abstract

This paper investigates the global topology of three-dimensional Hessian manifolds. We prove that every compact, orientable Hessian 3-manifold is either the Hantzsche Wendt manifold or admits the structure of a Kahler mapping torus. This result highlights a deep and intrinsic relationship between Hessian and Kahler geometries. Furthermore, we provide a classification of compact, orientable, three-dimensional Hessian manifolds.

The topology of 3-dimensional Hessian manifolds

Abstract

This paper investigates the global topology of three-dimensional Hessian manifolds. We prove that every compact, orientable Hessian 3-manifold is either the Hantzsche Wendt manifold or admits the structure of a Kahler mapping torus. This result highlights a deep and intrinsic relationship between Hessian and Kahler geometries. Furthermore, we provide a classification of compact, orientable, three-dimensional Hessian manifolds.
Paper Structure (17 sections, 13 theorems, 58 equations)

This paper contains 17 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Hessian manifolds and Kähler mapping tori
  3. Affine manifolds, Hessian metrics, and the first Koszul form
  4. Radiant Hessian manifold
  5. Hyperbolic affine manifold
  6. Kähler mapping torus
  7. Main results on compact Hessian $3$-manifolds
  8. Hessian manifold and Khaler mapping torus
  9. Case 1: $\alpha = 0$.
  10. Case 2: $\alpha \neq 0$.
  11. Betti numbers of 3-dimensional Hessian manifolds
  12. Remarks on Ricci curvature
  13. Hessian manifold and Seifert manifold
  14. Classification of 3-dimensional compact, orientable Hessian manifold
  15. General Case
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $(M, \nabla)$ be an locally flat manifold and $g$ a Riemannian metric. The following are equivalent:

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: shima1997geometryshima2007geometry
  • Proposition 2.1: shima1997geometry
  • Definition 2.4: koszul1961domainesshima1997geometry
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Koszul koszul1968deformations
  • Remark 2.9
  • ...and 26 more