Nonminimality of spirals in sub-Riemannian manifolds

TL;DR

This paper demonstrates that in certain analytic sub-Riemannian manifolds, spiral-like curves are not length minimizing near their centers, challenging assumptions about their optimality.

Contribution

It establishes the nonminimality of spiral-like curves in rank 2 sub-Riemannian manifolds under specific commutativity conditions, using a novel construction of competing curves.

Findings

Spiral-like curves are not length minimizing near the center in the specified manifolds.

The proof involves constructing a competing curve that shortens the spiral.

Results apply to analytic sub-Riemannian manifolds with rank 2 and commutativity conditions.

Abstract

We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.