Selfsimilar Hessian and conformally K\"ahler manifolds

TL;DR

This paper explores the geometric structures of Hessian manifolds, showing how their tangent bundles can be endowed with K"ahler and conformally K"ahler structures, especially under homogeneity and self-similarity conditions.

Contribution

It constructs homogeneous K"ahler and conformally K"ahler structures on tangent bundles of Hessian manifolds with symmetry and self-similarity properties.

Findings

Tangent bundles of homogeneous Hessian manifolds admit homogeneous K"ahler structures.

Selfsimilar Hessian manifolds with complete homothetic vector fields lead to conformally K"ahler structures on tangent bundles.

Group actions preserving Hessian structures induce rich geometric structures on tangent bundles.

Abstract

Let $(M,\nabla,g)$ be a Hessian manifold. Then the total space of the tangent bundle $TM$ can be endowed with a K\"ahler structure $\left(I,{\cal g}\right)$. We say that a homogeneous Hessian manifold is a Hessian manifold $(M,\nabla,g)$ endowed with a transitive action of a group $G$ preserving $\nabla$ and $g$. If $(M,\nabla,g)$ is a simply connected homogeneous Hessian manifold for a group $G$ then we construct an action of the group $G\ltimes_\theta \mathbb{R}^n$ on $TM=M\times \mathbb{R}^n$ such that $\left(TM,I,g\right)$ is a homogeneous K\"ahler manifold for the group $G\ltimes_\theta \mathbb{R}^n$. A selfsimilar Hessian manifold is a Hessian manifold endowed with a homothetic vector field $\xi$. Let $(M,\nabla,g,\xi)$ be a simply connected selfsimilar Hessian manifold such that $\xi$ is complete and $G$ be a group of automorphisms of $(M,\nabla,g,\xi)$ such that $G$ acts transitively on the level line ${g(\xi,\xi)=1}$. Then we construct homogeneous conformally K\"ahler structure on $TM$.