Selfsimilar Hessian and conformally K\"ahler manifolds

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 $TM$.