In case this is too general, here is a more specific question.

Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)?

## Background

A *Hessian manifold* is a Riemannian manifold which admits an atlas of coordinate charts whose transition maps are affine (i.e. $x \mapsto Ax+b$) and whose metric satisfies
$$ g_{ij} = \frac{\partial^2}{\partial x^i \partial x^j} \phi $$ in each coordinate chart for some potential function $\phi$ (which may depend on the chart). These spaces are also known as *affine Kahler manifolds*. If one prefers a coordinate-free definition, this is equivalent to a Riemannian manifold admitting a curvature- and torsion-free connection $D$ which satisfies $$D_X g(Y,Z) = D_Y g(X,Z) $$ for all vector fields $X,Y,Z$.

## Three preliminary examples

I'm aware of three examples of Hessian manifolds.

- If one considers a convex domain $\Omega \subset \mathbb{R}^n$ and a strongly convex potential $\phi: \Omega \to \mathbb{R}$, one can construct a Hessian manifold by setting $g = \frac{\partial^2}{\partial x^i \partial x^j} \phi$. In this case $\Omega$ serves as a global coordinate chart.
- The torus with its standard affine structure (i.e. $\mathbb{R}^n$ modulo a discrete lattice) can be made into a Hessian manifold. To do so, you can take a convex potential on $\mathbb{R}^n$ whose Hessian is invariant under the action of the discrete lattice.
- Less obviously, it is possible to another Hessian structure on the circle. For instance, if one considers the circle as the space $\mathbb{R}^+$ modulo the ''multiplicative lattice" $\{ 2^k | k \in \mathbb{Z} \}$, the resulting space is a Hessian manifold with the potential $-log (x)$ (defined on the affine universal cover).

There have been a considerable number of papers (and even a book) written on the geometry of Hessian manifolds. However, apart from these three examples (and their products), I have not been able to find any other examples in the literature. As such, I'm curious if there are other examples which are known, especially those which are compact.

## Some relevant results

Hessian manifolds manifold are necessarily affine, which greatly restricts their geometry. For instance, the fundamental group of a Hessian manifold must be infinite. However, many (most?) affine manifolds do not admit Hessian structures at all. Shima that the universal affine cover of a compact Hessian manifold is a convex domain [1]. As a result, the Hopf affine manifolds $\mathbb{S}^{n-1}\times \mathbb{S}^1$ do not admit Hessian metrics for $n>1$.

In dimension 2, it is possible to classify compact Hessian manifolds completely. The only compact Riemann surface which admits an affine structure (i.e. a flat connection) is the torus and there are 6 inequivalent affine structures. Yagi showed the only affine structures which admit Hessian metrics are products of the standard affine structure on the circle and the affine structure in Example 3 [2]. One can deform the convex potential (in fact, you can find a potential which induces an arbitrary analytic metric on some small neighborhood). However, the global structure of these spaces is fairly simple.

[1 ]*Shima, Hirohiko*, Hessian manifolds and convexity, Manifolds and Lie groups, Pap. in Honor of Y. Matsushima, Prog. Math. 14, 385-392 (1981). ZBL0481.53038.

[2] *Yagi, Katsumi*, On hessian structures on an affine manifold, Manifolds and Lie groups, Pap. in Honor of Y. Matsushima, Prog. Math. 14, 449-459 (1981). ZBL0495.53011.