Sub-Riemannian geometry and nonholonomic mechanics

Abstract

The purpose of the present paper is to study the geometry of a sub‐Riemannian manifold and to apply the results to the nonholonomic mechanical systems. First, we construct a linear connection on the horizontal distribution and obtain some Bianchi identities for it. Then, we introduce the horizontal sectional curvature, state a Schur Theorem for sub‐Riemannian geometry, and find a class of sub‐Riemannian manifolds of constant horizontal curvature. Also, we define the horizontal Ricci tensor and scalar curvature, and some sub‐Riemannian differential operators (gradient, divergence, Laplacian), extending some results from Riemannian geometry to the sub‐Riemannian setting. Finally, by using the sub‐Riemannian connection constructed here, we express the Lagrange‐d’ Alembert equations for a nonholonomic mechanical system in a form that is similar to the Newton’s equations from classical mechanics.