# American Institute of Mathematical Sciences

June  2016, 8(2): 139-167. doi: 10.3934/jgm.2016001

## Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

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

Received  September 2015 Revised  March 2016 Published  June 2016

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.
Citation: Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001
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##### References:
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