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On complexity and motion planning for co-rank one sub-riemannian metrics

Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the motion planning problem for generic sub-riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [10, 11]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C ∞ case, we study some non-generic generalizations in the analytic case.

On complexity and motion planning for co-rank one sub-Riemannian metrics

Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the motion planning problem for generic sub-Riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [CITE]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C∞ case, we study some non-generic generalizations in the analytic case.

On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz, Soheil Malekzadeh (2015)

Analysis and Geometry in Metric Spaces

We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

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

Davide Barilari, Luca Rizzi (2017)

Archivum Mathematicum

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.

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