Selfsimilar Hessian and conformally Kähler manifolds

2020/12/07 by Osipov, Pavel
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.03791

Abstract

Let (M,∇,g) be a Hessian manifold. Then the total space of the tangent bundle TM can be endowed with a Kähler structure (I,\cal g). We say that a homogeneous Hessian manifold is a Hessian manifold (M,∇,g) endowed with a transitive action of a group G preserving ∇ and g. If (M,∇,g) is a simply connected homogeneous Hessian manifold for a group G then we construct an action of the group G\ltimes_θℝⁿ on TM=M× ℝⁿ such that (TM,I,g) is a homogeneous Kähler manifold for the group G\ltimes_θℝⁿ. A selfsimilar Hessian manifold is a Hessian manifold endowed with a homothetic vector field ξ. Let (M,∇,g,ξ) be a simply connected selfsimilar Hessian manifold such that ξ is complete and G be a group of automorphisms of (M,∇,g,ξ) such that G acts transitively on the level line g(ξ,ξ)=1. Then we construct homogeneous conformally Kähler structure on TM.

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