## Journal of Generalized Lie Theory and Applications

### Locally Compact Homogeneous Spaces with Inner Metric

VN Berestovskii

#### 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.

#### Article information

Source
J. Gen. Lie Theory Appl., Volume 9, Number 1 (2015), 6 pages.

Dates
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.jglta/1443617963

Digital Object Identifier
doi:10.4172/1736-4337.1000223

Mathematical Reviews number (MathSciNet)
MR3624045

Zentralblatt MATH identifier
06499582

#### Citation

Berestovskii, VN. Locally Compact Homogeneous Spaces with Inner Metric. J. Gen. Lie Theory Appl. 9 (2015), no. 1, 6 pages. doi:10.4172/1736-4337.1000223. https://projecteuclid.org/euclid.jglta/1443617963