Journal of Generalized Lie Theory and Applications

Locally Compact Homogeneous Spaces with Inner Metric

VN Berestovskii

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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

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1443617963

Digital Object Identifier
doi:10.4172/1736-4337.1000223

Mathematical Reviews number (MathSciNet)
MR3624045

Zentralblatt MATH identifier
06499582

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

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


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