On the codimension of the abnormal set in step two Carnot groups
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 18

In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018002
Classification : 53C17, 22E25, 14M17
Keywords: Sard property, endpoint map, abnormal curves, Carnot groups, sub-Riemannian geometry

Ottazzi, Alessandro 1 ; Vittone, Davide 1

1 
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     title = {On the codimension of the abnormal set in step two {Carnot} groups},
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     year = {2019},
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Ottazzi, Alessandro; Vittone, Davide. On the codimension of the abnormal set in step two Carnot groups. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 18. doi: 10.1051/cocv/2018002

[1] A.A. Agrachev, Any sub-Riemannian metric has points of smoothness. Dokl. Akad. Nauk 424 (2009) 295–298 | Zbl | MR

[2] A.A. Agrachev, Some open problems, in Geometric Control Theory and Sub-Riemannian Geometry. Vol. 5 of Springer INdAM Series (2014) 1–14 | Zbl | MR

[3] A. Agrachev, A. Gentile and A. Lerario, Geodesics and horizontal-path spaces in Carnot groups. Geom. Topol. 19 (2015) 1569–1630 | Zbl | MR | DOI

[4] M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry. Vol. 144 of Progress in Mathematics. Birkhäuser, Basel (1996) 79-323 | Zbl | MR

[5] E. Le Donne, R. Montgomery, A. Ottazzi, P. Pansu and D. Vittone, Sard property for the endpoint map on some Carnot groups. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33 (2016) 1639–1666 | Zbl | MR | Numdam | DOI

[6] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Vol. 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2002) | Zbl | MR

[7] L. Rifford and E. Trélat, Morse-Sard type results in sub- Riemannian geometry. Math. Ann. 332 (2005) 145–159 | Zbl | MR | DOI

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