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 Sibirsk. Mat. Zh., 2018, Volume 59, Number 1, Pages 41–55 (Mi smj2952)

Geodesics and curvatures of special sub-Riemannian metrics on Lie groups

V. N. Berestovskiiab

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: Let $G$ be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space $M=G/K$ with the stabilizer $K$; $p\colon G\to G/K=M$ 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 $G=\operatorname{Sp}(1)\times\operatorname{Sp}(1)$ with the Riemannian symmetric space $S^3=\operatorname{Sp}(1)=G/\operatorname{diag}(\operatorname{Sp}(1)\times\operatorname{Sp}(1))$ the curvatures and torsions are calculated of images in $S^3$ 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.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.3087.2017/4.6 The author was supported by the Ministry of Education and Science of the Russian Federation (Grant 1.3087.2017/4.6).

DOI: https://doi.org/10.17377/smzh.2018.59.104

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English version:
Siberian Mathematical Journal, 2018, 59:1, 31–42

Bibliographic databases:

Document Type: Article
UDC: 514.752.8+514.763+514.765+514.764.227
MSC: 35R30

Citation: V. N. Berestovskii, “Geodesics and curvatures of special sub-Riemannian metrics on Lie groups”, Sibirsk. Mat. Zh., 59:1 (2018), 41–55; Siberian Math. J., 59:1 (2018), 31–42

Citation in format AMSBIB
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