Sub-Riemannian sphere in Martinet flat case

Abstract

Abstract. This article deals with the local sub-Riemannian g e ometry on R 3 (D g) w h e r e D is the distribution ker !, ! being the Martinet one-form: dz ; 1 2 y 2 dx and g is a Riemannian metric on D: We p r o ve that we can take g as a sum of squares adx 2 + cdy 2 : Then we analyze the at case where a = c = 1 : We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci, and the sub-Riemannian sphere. A direct consequence of our computations is to show that the sphere is not sub-analytic. Some of these computations are generalized to a one parameter deformation of the at case.

On the subRiemannian cut locus in a model of free two-step Carnot group

“…The case of h-type groups is discussed in [AM16]. Analysis in step three examples has been performed in [ABCK97], for the Martinet case, and in [AS11,AS15] for the Engel group. We also mention the very recent paper [BBN16], where a detailed discussion of the cut locus in the biHeisenberg group is performed.…”

Subriemannian geodesics of Carnot groups of step 3

“…This is impossible by Proposition 4.2, so ω ≡ 0. We have proved that if t ∈ [0, 1] is a Lebesgue point ofγ 1 , then…”

“…It suffices to prove that (4.11) holds for all Lebesgue points ofγ 1 . Letτ ∈ [0, 1] be a Lebesgue point ofγ 1 . Assume that λ 3 ā, [ā,γ 1 (τ) < 0 for somē a ∈ V 1 .…”

On the role of abnormal minimizers in sub-riemannian geometry

“…It will appear later that a good starting point to make the computations is to use the following normal form computed in [2] :…”

“…The abnormal line passing through 0 is given by γ : t −→ (±t, 0, 0). The computations in [2] show that we can make an additional normalization on the metric by taking either the restriction of a or c to the Martinet plane y = 0 equal to 0.…”