Self-similar solutions to the Hesse flow

TL;DR

This paper introduces Hesse solitons as self-similar solutions to the Hesse flow on Hessian manifolds, exploring their properties and connections to information geometry, including dual spaces and Einstein conditions.

Contribution

It defines Hesse solitons, analyzes their properties on proper Hessian manifolds, and links dual spaces to Hesse solitons, advancing understanding of geometric flows in information geometry.

Findings

Any compact proper Hesse soliton is expanding.

Non-trivial compact gradient Hesse solitons are proper.

Dual spaces of Hesse-Einstein manifolds are Hesse solitons.

Abstract

We define a Hesse soliton, that is, a self-similar solution to the Hesse flow on Hessian manifolds. On information geometry, the $e$-connection and the $m$-connection are important, which do not coincide with the Levi-Civita one. Therefore, it is interesting to consider a Hessian manifold with a flat connection which does not coincide with the Levi-Civita one. We call it a proper Hessian manifold. In this paper, we show that any compact proper Hesse soliton is expanding and any non-trivial compact gradient Hesse soliton is proper. Furthermore, we show that the dual space of a Hesse-Einstein manifold can be understood as a Hesse soliton.