Hybrid connections on Hessian manifolds

Abstract

We introduce a new class of affine connections on Hessian manifolds, called hybrid connections, characterized by the compatibility between their projective geometry with the underlying affine structure on the one hand and their infinitesimal holonomy with the Hessian metric on the other hand. In this paper, we investigate the properties of hybrid connections and prove that, on a given Hessian manifold, they are completely determined by the choice of a Hessian potential for the metric. In the special case of pseudo Euclidean manifolds, we identify canonical models and construct, in particular, a natural connection on the open unit ball that combines features of the Cayley Klein and Poincar\'e models of hyperbolic geometry. We also prove the existence and uniqueness (up to scaling) of a pseudo-Riemannian metric $h$ such that the geodesics of $\nabla$ admit parameterizations of constant speed with respect to $h$, which we call the isochrone metric.