Gradient horizontal de fonctions polynomiales

Đinh, Sĩ Tiệp; Kurdyka, Krzysztof; Orro, Patrice

doi:10.5802/aif.2481

Cited by 2 publications

(3 citation statements)

References 23 publications

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“…Proof The idea of proof is similar to Lemma 5.15 in [8]. By Proposition 3.2, for f ∈ L d , ∇ h f is a submersion on V f .…”

Section: Trajectories Of Horizontal Gradientmentioning

confidence: 91%

“…In [8], we have consider the trajectories of the horizontal gradient for a class of codimension one distribution, called splitting distribution, on R n . This class contains, in particular, contact structures.…”

mentioning

confidence: 99%

“…The lemma follows. [8], it follows that α(t) has a limit on ∂B. So from now on, assume that α(t) approaches to V f , i.e., there exists a sequence t m → t 0 < +∞, such that…”

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confidence: 99%

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Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4

Đinh

Kurdyka

2014

J Dyn Control Syst

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We investigate the set V f of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms ω 3 = dx 3 − x 1 dx 2 and ω 4 = dx 4 − x 3 dx 2 , endowed with the sub-Riemannian metric g SR = dx 2 1 + dx 2 2 . For a generic polynomial, we show that the set f of points in V f , where V f is not transverse to the Engel distribution, does not have a connected component which is contained in a fiber of f . Then, we prove that each trajectory of the horizontal gradient of f approaching the set V f has a limit.Keywords Engel structure · Horizontal gradient · Horizontal curve · Limit of trajectories Mathematics Subject Classifications (2010) 14P10 · 53C17 · 58A30 · 58Kxx IntroductionAn Engel structure is a non-integrable distribution of rank 2 on a four-dimensional manifold which satisfies the following conditions:where [., .] denotes the Lie bracket. Engel structures are stable (or generic) in the sense that all C 2 -small perturbation of an Engel structure is still an Engel structure. A manifold with an Engel structure is called an Engel manifold. In this paper, we will deal with the standard

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“…Proof The idea of proof is similar to Lemma 5.15 in [8]. By Proposition 3.2, for f ∈ L d , ∇ h f is a submersion on V f .…”

Section: Trajectories Of Horizontal Gradientmentioning

confidence: 91%

mentioning

confidence: 99%