On Jacobi fields and a canonical connection in sub-Riemannian geometry

Davide Barilari and Luca Rizzi

Address:
D. Barilari, Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR CNRS 7586, Université Paris-Diderot, Batiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France
L. Rizzi, CMAP École Polytechnique, Palaiseau and Équipe INRIA GECO Saclay Île-de-France Paris, France

E-mail:
davide.barilari@imj-prg.fr
luca.rizzi@cmap.polytechnique.fr

Abstract: In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [15]. We show why this connection is naturally nonlinear, and we discuss some of its properties.

AMSclassification: primary 53C17; secondary 53B21, 53B15.

Keywords: sub-Riemannian geometry, curvature, connection, Jacobi fields.

DOI: 10.5817/AM2017-2-77