Selfsimilar Hessian and conformally Kähler manifolds

英文摘要：

Let $(M,\nabla,g)$ be a Hessian manifold. Then the total space of the tangent bundle $TM$ can be endowed with a Kähler structure $\left(I,{\cal g}\right)$. 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 $G\ltimes_θ\mathbb{R}^n$ on $TM=M\times \mathbb{R}^n$ such that $\left(TM,I,g\right)$ is a homogeneous Kähler manifold for the group $G\ltimes_θ\mathbb{R}^n$. A selfsimilar Hessian manifold is a Hessian manifold endowed with a homothetic vector field $ξ$. Let $(M,\nabla,g,ξ)$ be a simply connected selfsimilar Hessian manifold such that $ξ$ is complete and $G$ be a group of automorphisms of $(M,\nabla,g,ξ)$ such that $G$ acts transitively on the level line ${g(ξ,ξ)=1}$. Then we construct homogeneous conformally Kähler structure on $TM$.