Convergence of the Hesse-Koszul flow on compact Hessian manifolds

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 ma…

Hesse manifolds and Hessian symmetries of multifield cosmological models

I give a brief overview of the mathematical theory of Noether symmetries of multifield cosmological models, which decompose naturally into visible and Hessian (a.k.a. 'hidden') symmetries. While visible symmetries correspond to those infinitesimal isometries of the Riemannian target space of the scalar field map which preserve the scalar potential, Hessian symmetries have a much deeper theory. The latter correspond to Hesse functions, defined as solutions of the so-called Hesse equation of the target space. By definition, a Hesse manifold is a Riemannian manifold which admits nontrivial Hesse …

Hess流形上几何流的孤子

On solitons of a geometric flow on Hessian manifolds

本文针对Hess流形上的几何流，引入Koszul孤子并研究其方程、性质与实例，通过关联仿射流形与复流形的方法拓展了几何问题的研究，该孤子在统计学与信息几何中具应用潜力。

三维Hessian流形的拓扑结构

The topology of 3-dimensional Hessian manifolds

本文研究了三维Hessian流形的拓扑结构，证明其为Hantzsche-Wendt流形或Kähler映射扭结构，并给出了完整的拓扑分类。

Hessian流形的Ricci曲率与Betti数

Ricci Curvature and Betti Numbers of Hessian Manifolds

研究Hessian度量在由第一Koszul形式生成的codimension-one foliation叶上的Ricci曲率性质，发现单叶非负Ricci曲率迫使Hessian度量为平坦，且给出Betti数的精确界。

Submanifolds in Koszul-Vinberg geometry

A Koszul-Vinberg manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our previous article [\textit{Contravariant Pseudo-Hessian manifolds and their associated Poisson structures}. \rm{Differential Geometry and its Applications} (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifold…