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Extremal Curves in Nilpotent Lie Groups


We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin maximum principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

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Correspondence to Enrico Le Donne.

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This work was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”, Padova.

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Le Donne, E., Leonardi, G.P., Monti, R. et al. Extremal Curves in Nilpotent Lie Groups. Geom. Funct. Anal. 23, 1371–1401 (2013).

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Keywords and phrases

  • Regularity of geodesics
  • abnormal curves
  • extremal curves
  • free nilpotent groups
  • Carnot groups
  • sub-Riemannian geometry
  • algebraic variety

Mathematics Subject Classification (2010)

  • 53C17
  • 49K30