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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.

  1. 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.
  2. 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.
  3. 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.

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    $\begingroup$ Aside: There are no known flat affine structures on closed hyperbolic 3-manifolds (or, for this matter, any closed 3-manifolds which are non-Seifert and non-Sol-manifolds). $\endgroup$ – Moishe Kohan Mar 10 at 13:29
  • $\begingroup$ Gabe K: I see that you have created (hessian-manifolds) tag. It might be useful to create also tag-wiki or at least tag-excerpt. It might help other users to use the tag correctly. (This is probably not a problem here, since the tag name seems to be descriptive enough.) Another reason is that the tags used on only one question are automatically deleted after certain time unless they have tag-wiki. $\endgroup$ – Martin Sleziak Mar 28 at 6:54
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Here is a collection of examples: Let $Q:\mathbb{R}^{n+1}\to\mathbb{R}$ be a (Lorentzian) quadratic form of type $(n,1)$, and let $L^+\subset\mathbb{R}^{n+1}$ be one component of the cone of time-like vectors with respect to $Q$ (i.e., where $Q$ is negative). Set $\phi = -\log(-Q)$, which is a well defined function on $L^+$, and let $g$ be the Hessian metric of $\phi$, i.e, for (any set of) linear coordinates $x^0,\ldots, x^n$, we have $$ g = \frac{\partial^2\phi}{\partial x^i\partial x^j}\,\mathrm{d}x^i\mathrm{d}x^j. $$ This $g$ is positive definite on $L^+$. Moreover, since $\phi$ is invariant under the group $\mathrm{O}(Q)$ and satisfies $\phi(rv) = \phi(v)-2\log r$ for any constant $r>0$, it follows that $g$ is invariant under the action of $\mathbb{R}^+{\cdot}\mathrm{O}(Q)$.

Now let $\Gamma\subset\mathrm{SO}(Q)$ be a discrete subgroup that acts freely and co-compactly on the $n$-dimensional level set $Q=-1$ in $L^+$ (which is a copy of hyperbolic space), so that the quotient of this level set by $\Gamma$ is a compact hyperbolic manifold. Also, let $\mathbb{Z}$ act on $L^+$ by $n\cdot v = r^n v$ for some real number $r>1$. The action generated by the commuting groups $\Gamma$ and $\mathbb{Z}$ is free and discrete and cocompact on $L^+$ and the metric $g$, which is obviously Hessian, is invariant under this action. Thus, $g$ descends to a Hessian metric on the compact quotient $M$ of $L^+$ by the action of $\Gamma\times\mathbb{Z}$.

Note that $M$ is diffeomorphic to the product of a circle and a compact hyperbolic $n$-manifold.

I imagine that there are other examples of this nature constructed on other invariant cones in vector spaces that have cocompact actions.

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  • $\begingroup$ Thanks! This is an excellent class of examples. If you don't mind me asking a follow-up question, in this example the function $\phi$ you defined induces a Hesse-Einstein metric (i.e., a Hessian metric whose tangent bundle is Kahler-Einstein). Are there examples like this where the Hesse-Einstein metric is harder to find? $\endgroup$ – Gabe K Mar 6 at 1:56
  • $\begingroup$ @GabeK: I don't know. However, maybe it's worth pointing out that, on these same underlying compact affine manifolds (hyperbolic manifolds crossed with the circle), it's easy to construct examples of Hessian metrics that are not Hesse-Einstein, since they are not rigid by any means. $\endgroup$ – Robert Bryant Mar 6 at 10:37

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