Open Access
Translator Disclaimer
2015 Locally Compact Homogeneous Spaces with Inner Metric
VN Berestovskii
Author Affiliations +
J. Gen. Lie Theory Appl. 9(1): 1-6 (2015). DOI: 10.4172/1736-4337.1000223

Abstract

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-) Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class Ω of all locally compact homogeneous spaces with inner metric is supplied with some metric $d_{BGH}$ such that 1) $(Ω, d_{BGH})$ is a complete metric space; 2) a sequences in $(Ω, d_{BGH})$ is converging if and only if it is converging in Gromov-Hausdor sense; 3) the subclasses M of homogeneous manifolds with inner metric and L G of connected Lie groups with leftinvariant Finslerian metric are everywhere dense in $(Ω, d_{BGH})$: It is given a metric characterization of Carnot groups with left-invariant sub-Finslerian metric. At the end are described homogeneous manifolds such that any invariant inner metric on any of them is Finslerian.

Citation

Download Citation

VN Berestovskii. "Locally Compact Homogeneous Spaces with Inner Metric." J. Gen. Lie Theory Appl. 9 (1) 1 - 6, 2015. https://doi.org/10.4172/1736-4337.1000223

Information

Published: 2015
First available in Project Euclid: 30 September 2015

zbMATH: 06499582
MathSciNet: MR3624045
Digital Object Identifier: 10.4172/1736-4337.1000223

Keywords: Carnot group , Cohn-Vossen theorem , Gromov-Haudor limit , Homogeneous (sub-)Finslerian manifold , Homogeneous (sub-)Riemannian manifold , Homogeneous isotropy irreducible space , Homogeneous manifold with inner metric , Homogeneous space with integrable invariant distributions , Lie algebra , Lie group , Locally compact homogeneous geodesic space , Non-holonomic metric geometry , Rashevsky-Chow theorem , Shortest arc , Submetry , Symmetric space , tangent cone

Rights: Copyright © 2015 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

JOURNAL ARTICLE
6 PAGES


SHARE
Vol.9 • No. 1 • 2015
Back to Top