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Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups. / Berestovskii, V. N.

In: Siberian Mathematical Journal, Vol. 59, No. 1, 01.01.2018, p. 31-42.

Research output: Contribution to journal › Article › peer-review

Harvard

Berestovskii, VN 2018, 'Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups', Siberian Mathematical Journal, vol. 59, no. 1, pp. 31-42. https://doi.org/10.1134/S0037446618010044

APA

Berestovskii, V. N. (2018). Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups. Siberian Mathematical Journal, 59(1), 31-42. https://doi.org/10.1134/S0037446618010044

Vancouver

Berestovskii VN. Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups. Siberian Mathematical Journal. 2018 Jan 1;59(1):31-42. doi: 10.1134/S0037446618010044

Author

Berestovskii, V. N. / Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups. In: Siberian Mathematical Journal. 2018 ; Vol. 59, No. 1. pp. 31-42.

BibTeX

@article{bcc550d01797454383b9b7267bf3e73b,
title = "Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups",
abstract = "Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.",
keywords = "geodesic orbit space, left invariant sub-Riemannian metric, Lie algebra, Lie group, normal geodesic, Riemannian symmetric space, DISTANCE, GROUP SO0(2,1), MANIFOLDS, GROUP SL(2)",
author = "Berestovskii, {V. N.}",
year = "2018",
month = jan,
day = "1",
doi = "10.1134/S0037446618010044",
language = "English",
volume = "59",
pages = "31--42",
journal = "Siberian Mathematical Journal",
issn = "0037-4466",
publisher = "MAIK NAUKA/INTERPERIODICA/SPRINGER",
number = "1",

}

RIS

TY - JOUR

T1 - Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

AU - Berestovskii, V. N.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.

AB - Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.

KW - geodesic orbit space

KW - left invariant sub-Riemannian metric

KW - Lie algebra

KW - Lie group

KW - normal geodesic

KW - Riemannian symmetric space

KW - DISTANCE

KW - GROUP SO0(2,1)

KW - MANIFOLDS

KW - GROUP SL(2)

UR - http://www.scopus.com/inward/record.url?scp=85043522149&partnerID=8YFLogxK

U2 - 10.1134/S0037446618010044

DO - 10.1134/S0037446618010044

M3 - Article

AN - SCOPUS:85043522149

VL - 59

SP - 31

EP - 42

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 1

ER -

ID: 12099592