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


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.


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VN Berestovskii. "Locally Compact Homogeneous Spaces with Inner Metric." J. Gen. Lie Theory Appl. 9 (1) 1 - 6, 2015.


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)

Vol.9 • No. 1 • 2015
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