Selfsimilar Hessian and conformally K\"ahler manifolds

DOI10.1007/S10455-022-09861-1arXiv2012.03791MaRDI QIDQ6355476

Pavel Osipov

Publication date: 7 December 2020

Abstract: Let $(M,abla,g)$ be a Hessian manifold. Then the total space of the tangent bundle $TM$ can be endowed with a K"ahler structure $left(I,calgight)$. We say that a homogeneous Hessian manifold is a Hessian manifold $(M,abla,g)$ endowed with a transitive action of a group $G$ preserving $abla$ and $g$. If $(M,abla,g)$ is a simply connected homogeneous Hessian manifold for a group $G$ then we construct an action of the group $Gltime{s}_{h}etamathbb{R}^n$ on $TM=Mimesmathbb{R}^n$ such that $left(TM,I,gight)$ is a homogeneous K"ahler manifold for the group $Gltime{s}_{h}etamathbb{R}^n$. A selfsimilar Hessian manifold is a Hessian manifold endowed with a homothetic vector field $xi$. Let $(M,abla,g,xi)$ be a simply connected selfsimilar Hessian manifold such that $xi$ is complete and $G$ be a group of automorphisms of $(M,abla,g,xi)$ such that $G$ acts transitively on the level line $g(xi,xi)=1$. Then we construct homogeneous conformally K"ahler structure on $TM$.

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Kähler manifolds (32Q15) Hyper-Kähler and quaternionic Kähler geometry, ``special geometry (53C26)

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