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A Hessian manifold (also known as an ``affine-Kähler manifold") is a Riemannian manifold which admits a curvature- and torsion-free affine connection $D$ (i.e., an affine manifold) and whose Riemannian metric $g$ is locally given by the Hessian of a potential function $\Phi$. In other words, we have that $g = D^2 \Phi$.

There is equivalent definition for Hessian manifolds in terms of dually flat connections. More precisely, a Hessian manifold is a Riemannian manifold $(M,g)$ with two flat connections $D$ and $D^*$ satisfying \begin{equation} \label{conjugateconnection} \mathcal{X}(g(\mathcal{Y},\mathcal{Z})) = g(D_\mathcal{X} \mathcal{Y}, \mathcal{Z}) + g(\mathcal{Y}, D^*_\mathcal{X} \mathcal{Z}) \end{equation} for all vector fields $\mathcal{X},~ \mathcal{Y}$, and $\mathcal{Z}$.

Hessian manifolds have many special properties, a few of which are listed below.

1. For every Hessian manifold, there is a dual Hessian manifold induced by the dual connection $D^\ast$ and whose potentials are given by the Legendre transformations of the original potentials. These manifolds are isometric as Riemannian manifolds, but their affine structures can be distinct.
2. The tangent bundle of a Hessian manifold, when induced with the Sasaki metric, is a Kahler manifold.
3. The affine universal cover of a Hessian manifold is a convex domain in affine space.
4. In the context of information geometry, Hessian manifolds can be constructed by exponential families by using the Fisher metric and the natural parameters to induce a connection.

A standard reference on the topic is The Geometry of Hessian Structures by Hirohiko Shima. Another reference is the paper ``Affine Spheres and Kähler-Einstein Metrics" by John Loftin which discusses some more analytic aspects of the theory.