Self-similar solutions to the Hesse flow

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.