Subriemannian geodesics of Carnot groups of step 3

Tan, Kang-Hai; Yang, Xiaoping

doi:10.1051/cocv/2012006

Cited by 9 publications

(13 citation statements)

References 23 publications

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“…The results here are less explicit because they involve a "lifting" procedure from a nonfree group to a free one, see Theorem 5.4. However, the results are precise enough to deduce in a purely algebraic way some interesting facts on Carnot-Carathéodory geodesics, such as the C ∞ smoothness of length minimizing curves in Carnot groups of step 3 (see [13]), and Golè-Karidi's example [5] of a strictly abnormal extremal in a Carnot group. These and other examples are briefly discussed in Section 6.…”

Section: Introductionmentioning

confidence: 99%

Extremal Curves in Nilpotent Lie Groups

et al. 2013

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Section: Introductionmentioning

confidence: 99%

Extremal Curves in Nilpotent Lie Groups

et al. 2013

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“…[AS95,ABB17], thus every length-minimizer admits a normal lift, Date: December 5, 2018. and is hence smooth. For step 3 structures, the situation is already more complicated and a positive answer to the above problem is known only for Carnot groups (where, actually, length-minimizers are proved to be C ∞ ), see [LDLMV13,TY13]. When the sub-Riemannian structure is analytic, more is known on the size of the set of points where a length-minimizer can lose regularity [Sus14], regardless of the rank and of the step of the distribution.…”

Section: Introductionmentioning

confidence: 99%

On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures

Barilari

Chitour

Jean

et al. 2020

Journal de Mathématiques Pures et Appliquées

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We prove the C 1 regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class C 1 . arXiv:1804.00971v2 [math.OC] 4 Dec 2018If the sub-Riemannian manifold has rank 2 and step at most 4, the assumption in Theorem 1 is trivially satisfied by every abnormal minimizer γ and we immediately obtain the following corollary.Corollary 2. Assume that the sub-Riemannian structure has rank 2 and step at most 4. Then all length-minimizers are of class C 1 .It is legitimate to ask whether the C 1 regularity in the Open Problem can be further improved. Indeed, the argument behind our proof permits to obtain C ∞ regularity of length-minimizers under an additional nilpotency condition on the Lie algebra generated by horizontal vector fields.Proposition 3. Assume that D is generated by two vector fields X 1 , X 2 such that the Lie algebra Lie{X 1 , X 2 } is nilpotent of step at most 4. Then for every sub-Riemannian structure (D, g) on M , the corresponding length-minimizers are of class C ∞ .The above proposition applies in particular to Carnot groups of rank 2 and step at most 4. In this case we recover the results obtained in [LM08, Example 4.6].The strategy of proof of Theorem 1 is to show that, at points where they are not of class C 1 , length-minimizers can admit only corner-like singularities. This is done by a careful asymptotic analysis of the differential equations satisfied by the abnormal lift, which exploits their Hamiltonian structure. We can then conclude thanks to the following result. Theorem 4 ([HL16]). Let M be a sub-Riemannian manifold. Let T > 0 and let γ : [−T, T ] → M be a horizontal curve parametrized by arclength. Assume that, in local

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“…Our result uses ideas of Tan and Yang [TY13] and the fact that in an arbitrary polarized Lie group the Sard Property holds for normal-abnormal curves, see Lemma 2.32. Theorem 1.5.…”

Section: Introductionmentioning

confidence: 99%

“…Sard Property for abnormal length minimizers. In [TY13] Tan and Yang proved that in sub-Riemannian step-3 Carnot groups all length minimizing curves are smooth. They also claim that in this setting all abnormal length minimizing curves are normal.…”

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Sard property for the endpoint map on some Carnot groups

Donne

Montgomery

Ottazzi

et al. 2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. In Carnot-Carathéodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

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Subriemannian geodesics of Carnot groups of step 3

Cited by 9 publications

References 23 publications

Extremal Curves in Nilpotent Lie Groups

Extremal Curves in Nilpotent Lie Groups

On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures

Sard property for the endpoint map on some Carnot groups

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