A compact Hessian manifold is a compact affine manifold (a manifold with a flat, torsion-free connection on its tangent bundle) carrying a Riemannian metric that is locally the Hessian of a potential function in affine coordinates. Liu (2025) proved that every compact Hessian manifold of dimension at most six is finitely covered by a product of a flat torus and a compact hyperbolic affine manifold, with the metric splitting accordingly; the proof depends on Auslander-conjecture-type results of Abels–Margulis–Soifer that are available only through dimension six, and those remain open in dimensions at least seven. The problem asks whether this finite-cover product splitting holds for every compact Hessian manifold, in particular in dimensions at least seven, where the parallel-volume input used in the proof of the splitting theorem is unavailable. The question is settled by either a proof of the splitting in full generality or a compact Hessian manifold that admits no such finite-cover decomposition.

Background

An affine manifold is a pair (M,∇)(M,\nabla) where ∇\nabla is a flat, torsion-free connection on the tangent bundle of MM; equivalently, MM is locally modeled on affine space with transition maps in the affine group Rn⋊GLn(R)\mathbb{R}^n\rtimes GL_n(\mathbb{R}). Integrating the flat connection gives the affine development, a local diffeomorphism from the universal cover M~\widetilde{M} to Rn\mathbb{R}^n. A Riemannian metric hh on an affine manifold is a Hessian metric if it is locally of the form h=∇(df)h=\nabla(df) for a smooth potential function ff in affine coordinates, a condition equivalent to the total symmetry of the tensor ∇h\nabla h; the pair ((M,∇),h)((M,\nabla),h) is then a Hessian manifold. Hessian manifolds arise in information geometry, in the theory of special Lagrangian fibrations, and as quotients of convex cones. Flat Riemannian manifolds (compact Bieberbach manifolds, i.e. quotients Rn/Γ\mathbb{R}^n/\Gamma with Γ\Gamma acting freely, properly discontinuously and cocompactly by Euclidean isometries) are the basic examples, and quotients of regular convex domains by cocompact discrete groups of affine automorphisms provide non-flat ones. An affine manifold is called hyperbolic if the affine development of its universal cover is a diffeomorphism onto a regular convex domain, that is, a convex domain containing no complete affine line.

Two theorems of Shima and Yagi (1997) anchor the structure theory of compact Hessian manifolds. First, the affine development of the universal cover of a complete Hessian manifold is a diffeamorphism onto a convex domain. Second, for an orientable compact Hessian manifold the following are equivalent: the first Koszul form (the canonical closed 1-form obtained by tracing the covariant derivative of the metric) vanishes; ∇\nabla is the Levi-Civita connection of hh; and MM admits a parallel volume form, meaning a nowhere-vanishing top-degree form Ω\Omega with ∇Ω=0\nabla\Omega=0. In that case the metric is flat Riemannian.

Chern (1955) conjectured that every compact affine manifold has vanishing Euler characteristic. This is known for surfaces (Milnor), for compact complete affine manifolds (Kostant–Sullivan), and for special affine manifolds, those carrying a parallel volume form (Klingler). Liu (2025) confirmed it for all compact Hessian manifolds, in every dimension, by showing that every compact orientable Hessian manifold is either flat Riemannian or fibers over the circle. The same paper established the structural theorem at issue here: every compact Hessian manifold of dimension n≤6n\le 6 admits a finite cover that splits, simultaneously as an affine manifold and as a Riemannian manifold, into the product of a flat torus and a compact hyperbolic affine manifold carrying a Hessian metric. The hypothesis n≤6n\le 6 enters the proof at a single point: a theorem of Abels, Margulis and Soifer (2002) implies that a compact complete affine manifold of dimension at most six carries a parallel volume form, which by the Shima–Yagi equivalence forces the affine fibers of the development to be flat. That input is an instance of the Auslander conjecture (1964): the fundamental group of a compact complete affine manifold, a crystallographic subgroup of the affine group, is virtually solvable, i.e. contains a solvable subgroup of finite index. The Auslander conjecture is proved for dimension at most three by Fried and Goldman and for dimensions below seven by Abels, Margulis and Soifer, and it remains open in dimensions at least seven, so the only known route to the splitting theorem breaks down exactly where the theorem stops.

Problem Statement

Theorem 4.15 of Liu (2025) states: let ((M,∇),h)((M,\nabla),h) be a compact Hessian manifold of dimension n≤6n\le 6. Then there exist a finite covering map f:E→Mf:E\to M and a compact hyperbolic affine manifold (B,∇B)(B,\nabla_B) of dimension k≤nk\le n such that

(E,f∗∇)=(B,∇B)×(Tn−k,D),f∗h=g⊕δ,(E,f^*\nabla)=(B,\nabla_B)\times(\mathbb{T}^{n-k},D),\qquad f^*h=g\oplus\delta,

where DD is the flat Levi-Civita connection of the flat torus (Tn−k,δ)(\mathbb{T}^{n-k},\delta), and gg is a Hessian metric on (B,∇B)(B,\nabla_B).

Question: does the same conclusion hold for every compact Hessian manifold, without the dimension restriction - in particular in every dimension n≥7n\ge 7, where the parallel-volume input used in the proof of the splitting theorem is unavailable? The question is settled by either a proof of the splitting in all dimensions or a compact Hessian manifold (of some dimension at least seven) that admits no finite covering splitting, as an affine manifold and as a Riemannian product, into the product of a flat torus and a compact hyperbolic affine manifold carrying a Hessian metric.

References

  1. Hanwen Liu, On Topology of Compact Hessian Manifolds, arXiv:2509.01176 (2025; v7 June 2026), doi:10.48550/arXiv.2509.01176, https://arxiv.org/abs/2509.01176
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