arXiv subjects

Pavel Osipov

Publications and source records attributed to Pavel Osipov.

6 recordsLinked to original sources

arXiv2026

Kunneth formula for Hessian manifolds

We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a Künneth formula \[ H^{p,q}(M\times N) \cong \bigoplus_{i,j} H^{i,j}(M)\otimes H^{p-i,q-j}(N) \] for flat affine manifolds $M,N$ with at least one compact. For compact manifolds we also give a proof via Hodge theory on flat affine manifolds, analogous to the classical Künneth formula for Dolbeault cohomology. We apply this formula to Hessian manifolds. A Hessian metric $g$ defines a class $[g]\in H^{1,1}(M)$, and metrics in the same class differ by $Dα$ for a closed $1$-form $α$. Using the Künneth formula we describe all Hessian metrics on products, on products with hyperbolic manifolds, and on manifolds admitting a flat Riemannian metric.

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arXiv2023

Locally conformally Hessian and statistical manifolds

A statistical manifold $\left(M,D,g\right)$ is a manifold $M$ endowed with a torsion-free connection $D$ and a Riemannian metric $g$ such that the tensor $D g$ is totally symmetric. If $D$ is flat then $\left(M,g,D\right)$ is a Hessian manifold. A locally conformally Hessian (l.c.H) manifold is a quotient of a Hessian manifold $(C,\nabla,g)$ such that the monodromy group acts on $C$ by Hessian homotheties, i.e. this action preserves $\nabla$ and multiplies $g$ by a group character. The l.c.H. rank is the rank of the image of this character considered as a function from the monodromy group to real numbers. A l.c.H. manifold is called radiant if the Lee vector field $ξ$ is Killing and satisfies $\nabla ξ=λ\Id$. We prove that the set of radiant l.c.H. metrics of l.c.H. rank 1 is dense in the set of all radiant l.c.H. metrics. We prove a structure theorem for compact radiant l.c.H. manifold of l.c.H. rank 1. Every such manifold $C$ is fibered over a circle, the fibers are statistical manifolds of constant curvature, the fibration is locally trivial, and $C$ is reconstructed from the statistical structure on the fibers and the monodromy automorphism induced by this fibration.

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arXiv2022

Statistical Lie algebras of a constant curvature and locally conformally Kähler Lie algebras

We show that a statistical manifold manifold of a constant non-zero curvature can be realised as a level line of Hessian potential on a Hessian cone. We construct a Sasakian structure on $TM\times\R$ by a statistical manifold manifold of a constant non-zero curvature on $M$. By a statistical Lie algebra of a constant non-zero Lie algebra we construct a l.c.K Lie algebra.

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arXiv2021

Selfsimilar Hessian manifolds

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian manifold or a Eucledean space. A radiant Hessian manifold is selfsimilar Hessian manifold $\left(M,\nabla,g,ξ\right)$ such that $\nablaξ=λ\text{Id}$. We prove that any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product radiant Hessian manifolds and describe the local structure of radiant selfsimialar Hessian manifolds.

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arXiv2021

Selfsimilar Hessian and conformally Kähler manifolds

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 $TM$.

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arXiv2019

Projective Hessian and Sasakian manifolds

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry is an odd-dimensional counterpart of the Kähler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous to the one between Hessian and Kähler manifolds. In particular, we construct a Sasakian structure on $TM\times \mathbb{R}$ from a projective Hessian structure on $M$. Especially, we are interested in the case of invariant structure on Lie groups. We define semi-Sasakian Lie groups as a generalization of Sasakian Lie groups. Then we construct a semi-Sasakian structure on a group $G\ltimes \mathbb{R}^{n+1}$ for a projective Hessian Lie group $G$. Further, we describe examples of homogeneous Hessian Lie groups and corresponding semi-Sasakian Lie groups. The big class of projective Hessian Lie groups can be constructed by homogeneous regular domains in $\mathbb{R}^n$. The groups $\text{SO}(2)$ and $\text{SU}(2)$ belong to another kind of examples. Using them, we construct semi-Sasakian structures on the group of the Euclidean motions of the real plane and the group of isometries of the complex plane.

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