Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

Dennis I.  Barrett; Rory  Biggs; Claudiu C.  Remsing; Olga  Rossi

doi:10.3934/jgm.2016001

Article Contents

2016, Volume 8, Issue 2: 139-167. Doi: 10.3934/jgm.2016001

This issue Previous Article Next Article Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

1.
Department of Mathematics (Pure and Applied), Rhodes University, 6140 Grahamstown
2.
Department of Mathematics, Faculty of Science, University of Ostrava, 701 03 Ostrava, Czech Republic and Department of Mathematics, Ghent University, B-9000 Ghent

Received: August 31, 2015

Revised: February 29, 2016

Published: May 2016

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Abstract

We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classification naturally splits into two cases. In the first case, it reduces to a classification of left-invariant sub-Riemannian structures. In the second case, we find a canonical frame with which to directly compare equivalence classes.

Keywords:

Nonholonomic mechanical system,
invariant nonholonomic Riemannian structure,
isometric invariant,
nonholonomic connection,
Schouten curvature tensor.

Mathematics Subject Classification: Primary: 70G45, 37J60; Secondary: 53A55, 22E30.

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