Locally Compact Homogeneous Spaces with Inner Metric
Sobolev Institute of Mathematics SB RAS, Acad. Koptyug Avenue, 4 630090, Novosibirsk, Russia
- Corresponding Author:
- Berestovskii VN
Sobolev Institute of Mathematics
SB RAS, Acad. Koptyug Avenue
4630090, Novosibirsk, Russia
E-mail: [email protected]
Received date: January 05, 2015 Accepted date: June 29, 2015 Published date: July 07, 2015
Citation: Berestovskii VN (2015) Locally Compact Homogeneous Spaces with Inner Metric. J Generalized Lie Theory Appl 9:223. doi:10.4172/1736-4337.1000223
Copyright: © 2015 Berestovskii VN. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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 dBGH such that 1) (Ω, dBGH) is a complete metric space; 2) a sequences in (Ω, dBGH) 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, dBGH): 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.