On a problem of A. M. Vasil'ev
type: Item
instance of: scholarly article
title: On a problem of A. M. Vasil'ev
zbMATH Open document ID: 0834.53039
DOI: 10.1007/BF02336540
published in: Lithuanian Mathematical Journal
publication date: 1995-11-07T00:00:00Z
review text: Let \(G\) be a connected Lie group and \(H\) a closed subgroup of \(G\). We assume that the quotient space \(M = G/H\) is reductive. Reductive spaces belong to the class of homogeneous spaces having canonical invariant affine connections. These connections have a characteristic property: geodesics of these connections are trajectories of one-parameter subgroups of motion groups of the homogeneous space. Invariant affine connections of the reductive homogeneous space \(M = G/H\) having the above mentioned property will be called \(G\)-connections. The problem of A. M. Vasil'ev is to find invariant affine connections of the reductive homogeneous space \(M = G/H\), the geodesics of which are trajectories of one-parameter subgroups of normal extensions \({\mathcal G}\) of the Lie group \(G\) of their motions. Such connections will be called \(\Gamma\)- connections. Every \(G\)-connection is a \(\Gamma\)-connection. This is a simplified solution to the Vasil'ev problem. It may turn out that the Vasil'ev problem can be solved using its simplified variant. For this, it is necessary and sufficient that each \(\Gamma\)-connection of \(M = G/H\) be a \({\mathcal G}\)-connection of \(M = {\mathcal G}/{\mathcal H}\). This allows us to formulate a new problem: Is it true that each Vasil'ev problem is solved with the help of its simplified variant? In this paper, the author proves that the above problem is nontrivial and gives a partial solution to it. In particular, he proves that semisimple manifolds admit only a trivial solution to the Vasil'ev problem.
reviewed by: Zixin Hou
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 813150
zbMATH Keywords: invariant affine connections
zbMATH Keywords: reductive homogeneous space
zbMATH Keywords: Vasil'ev problem
author: R. V. Vosylius
MaRDI profile type: Publication
cites work: Invariant Affine Connections on Homogeneous Spaces
description: scientific article; zbMATH DE number 813150