Hybrid connections on Hessian manifolds

A Hessian manifold $(M,D,g)$ is a manifold $M$ with a flat connection $D$ and a Riemannian or pseudo-Riemannian metric $g$ that is locally of the form $D^{2}f$ for some function $f$. On a Hessian manifold $(M,D,g)$, we define a hybrid connection as an incompressible affine connection $\mathrm{\nabla }$ that is projectively flat relative to $D$ (its unparametrized geodesics are aligned with the affine structure of $D$) and whose first-order infinitesimal holonomy at each point of $M$ is an infinitesimal isometry of the pseudo-Riemannian metric $g$. In this paper, we investigate the properties of hybrid connections, proving in particular that for a hybrid connection $\mathrm{\nabla }$, the difference $\mathrm{\nabla }-D$ is determined by the logarithmic differential of a function that serves as a Hessian potential for $g$. In the special case of pseudo-Euclidean manifolds, we identify canonical models and obtain in particular a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincaré hyperbolic models. We also find a unique (up to a scaling) pseudo-Riemannian metric $h$ such that unparameterized geodesics of $\mathrm{\nabla }$ have a constant speed with respect to the so-called isochrone metric $h$.