Selfsimilar Hessian and conformally K\"ahler manifolds

Let $(M,\mathrm{\nabla },g)$ be a Hessian manifold. Then the total space of the tangent bundle $TM$ can be endowed with a K"ahler structure $(I,\mathcal{g})$. We say that a homogeneous Hessian manifold is a Hessian manifold $(M,\mathrm{\nabla },g)$ endowed with a transitive action of a group $G$ preserving $\mathrm{\nabla }$ and $g$. If $(M,\mathrm{\nabla },g)$ 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 $(TM,I,g)$ is a homogeneous K"ahler 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 $(M,\mathrm{\nabla },g,\xi )$ be a simply connected selfsimilar Hessian manifold such that $\xi$ is complete and $G$ be a group of automorphisms of $(M,\mathrm{\nabla },g,\xi )$ such that $G$ acts transitively on the level line $g(\xi ,\xi )=1$. Then we construct homogeneous conformally K"ahler structure on $TM$.