Homogeneous manifolds with intrinsic metric. II
type: Item
instance of: scholarly article
title: Homogeneous manifolds with intrinsic metric. II
zbMATH Open document ID: 0681.53029
DOI: 10.1007/BF00971372
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: [For part I, cf. Sib. Math. J. 29, No.6, 887-897 (1988); translation from Sib. Mat. Zh. 29, No.6(172), 17-29 (1988; Zbl 0671.53036).] This paper completes the investigation of the author in the domain of the metric structure of homogeneous manifolds with intrinsic metric. The main result is: the homogeneous spaces with intrinsic metric are exactly factor-spaces G/H of connected Lie groups G by their compact subgroups H equipped with an (invariant with respect to the canonical action G on G/H) Carnot-\(Caratheodory\)-\(Finsler\) metric. Such metric \(d_ c\) is given by a ``completely nonholonomic'' G-invariant distribution \(\Delta\) on \(M=G/H\) and by the G-invariant norm \(F=F(p,\cdot)\), \(p\in G/H\) on the linear subspace \(\Delta\) (p) of the tangent space \(M_ p\) to M in the point p. A necessary and sufficient condition that the G-invariant intrinsic metric on G/H is of Finsler type is found and is given in terms of the corresponding Lie algebras. Homogeneous Riemannian and Finsler manifolds are characterized and described.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C70
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4115459
zbMATH Keywords: homogeneous manifolds
zbMATH Keywords: intrinsic metric
zbMATH Keywords: Lie groups
zbMATH Keywords: Carnot- \(Caratheodory\)-\(Finsler\) metric
zbMATH Keywords: invariant distribution
author: V. N. Berestovskij
reviewed by: A. Fleischer
MaRDI profile type: Publication
cites work: Critical theories
cites work: Q5797644
cites work: Q5651882
cites work: Q5734920
cites work: The fundamental equations of a submersion
cites work: Q3040034
cites work: Q5719719
cites work: Q5627393
cites work: Q5686690
cites work: Q4746404
cites work: Cohomology Theory of Lie Groups and Lie Algebras
cites work: Q3242599
full work available at URL: https://doi.org/10.1007/bf00971372
OpenAlex ID: W1974334230
description: scientific article; zbMATH DE number 4115459