Shortest and straightest geodesics in sub-Riemannian geometry

JOURNAL OF GEOMETRY AND PHYSICS

卷:155

Article

Alekseevsky, Dmitri^1,2 

[1] AA Kharkevich Inst Informat Transmiss Problems, B Karetnuj Per 19, Moscow 127051, Russia

[2] Univ Hradec Kralove, Fac Sci, Rokitanskeho 62, Hradec Kralove 50003, Czech Republic

关键词: sub-Riemannian geometry;    sub-Riemannian geodesics;    sub-Riemannian homogeneolus manifolds;   

DOI  :  10.1016/j.geomphys.2020.103713

来源: Elsevier

【 摘 要 】

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and generalization of sub-Riemannian geodesics and discuss interrelations between different definitions. For Chaplygin transversally homogeneous sub-Riemannian manifold Q, we prove that straightest geodesics (defined as geodesics of the Schouten partial connection) coincide with shortest geodesics (defined as the projection to Q of integral curves (with trivial initial covector) of the sub-Riemannian Hamiltonian system). This gives a Hamiltonization of Chaplygin systems in non-holonomic mechanics. We consider a class of homogeneous sub-Riemannian manifolds, where straightest geodesics coincide with shortest geodesics, and give a description of all sub-Riemannian symmetric spaces in terms of affine symmetric spaces. (C) 2020 Published by Elsevier B.V.

【 授权许可】

Free