Invariant intrinsic Finsler metrics on homogeneous spaces and strong subalgebras of Lie algebras

Горбацевич, Владимир Витальевич

doi:10.1007/s11202-008-0004-1

Cited by 15 publications

(5 citation statements)

References 9 publications

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“…Gorbatsevich [8,44] studied general homogeneous spaces with connected stabilizer subgroup from the class HMIID in detail. He described corresponding transitive Lie groups and stabilizer subgroups in the case when the transitive group is semisimple or solvable, and partly, in the case of general transitive Lie groups.…”

Section: Theorem 23 Let Assume That M=g/h (Where G Is a Connected LImentioning

confidence: 99%

Locally Compact Homogeneous Spaces with Inner Metric

2015

J Generalized Lie Theory Appl

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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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Section: Theorem 23 Let Assume That M=g/h (Where G Is a Connected LImentioning

confidence: 99%

Locally Compact Homogeneous Spaces with Inner Metric

2015

J Generalized Lie Theory Appl

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“…V. V. Gorbatsevich [44], [8] studied general homogeneous spaces with connected stabilizer subgroup from the class HMIID in detail. He described corresponding transitive Lie groups and stabilizer subgroups in the case when the transitive group is semisimple or solvable, and partly, in the case of general transitive Lie groups.…”

Section: Homogeneous Manifolds With Integrable Invariant Distributionsmentioning

confidence: 99%

Locally compact homogeneous spaces with inner metric

Berestovskiî

2014

Preprint

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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-Hausdorff sense; 3) the subclasses M of homogeneous manifolds with inner metric and LG of connected Lie groups with left-invariant 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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“…All isotropy irreducible homogeneous spaces G/H belong to the class of homogeneous spaces with integrable invariant distributions which was introduced in the author's article [28]. The articles [29,30] contain more incomplete but sufficiently advanced results about their structure and classification. Szabo's proof [17] gives a positive answer to Wolf's question in [4]: Is it possible to prove that the two-point homogeneous Riemannian spaces are symmetric without using the classification theorems?…”

Section: Lemmamentioning

confidence: 99%