1999/11/26 by Paolo Piccione, Piccione, Paolo, Daniel V. Tausk +1
Mathematics · Physics and Astronomy · #37J05 #37J50 #37J60 #53C17 #70H03 #70H05 #Advanced Differential Geometry Research #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:37J05 #msc:37J50 #msc:37J60 #msc:53C17 #msc:70H03 #msc:70H05
paper · pdf · doi:10.48550/arxiv.math/9911215
LaTeX2e, amsart class, 25 pages Replacement on Jan 5th 2000: added Appendix B Replacement on May 23rd 2000: modified Abstract and Introduction
openalex publication_date 1999/11/26 · arxiv created 2000/05/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds \mathcal P and \mathcal Q of a manifold \mathcal M endowed with a distribution \mathcal D⊂ T\M. We give a different proof, that holds in a more general context, of a result by Bismut (Large Deviations and the Malliavin Calculus, Birkhauser, 1984) stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functional. We use the Lagrangian multipliers method in a Hilbert manifold setting, which leads to a characterization of the abnormal extremizers (critical points of the endpoint map) as curves where the linear constraint fails to be regular. Finally, we describe a modification of a result by Liu and Sussmann that shows the global distance minimizing property of sufficiently small portions of normal extremizers between a point and a submanifold.