Mathematische Annalen

, Volume 332, Issue 1, pp 145–159 | Cite as

Morse-Sard type results in sub-Riemannian geometry

  • L. Rifford
  • E. TrélatEmail author


Let (M,Δ,g) be a sub-Riemannian manifold and x0M. Assuming that Chow’s condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution Δ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M, and in the case of corank one is of full Lebesgue measure in M.


Manifold Lebesgue Measure Dense Subset Exponential Mapping Full Lebesgue Measure 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Equipe d’Analyse Numérique et EDP, UMR 8628Université Paris-SudOrsay CedexFrance

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