Selfsimilar Hessian and conformally Kähler manifolds

Abstract

Let $(M,\nabla ,g)$ be a Hessian manifold. Then the total space of the tangent bundle $TM$ can be endowed with a Kähler structure $\left(I,g\right)$. 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 $G{⋉}_{\theta }{\mathbb{R}}^n$ on $TM=M\times {\mathbb{R}}^n$ such that $\left(TM,I,g\right)$ is a homogeneous Kähler manifold for the group $G{⋉}_{\theta }{\mathbb{R}}^n$. 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 Kähler structure on $TM$.