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Measures of transverse paths in sub-Riemannian geometry

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Abstract

We define a class of lengths of paths in a sub-Riemannian manifold. It includes the length of horizontal paths but also measures the length of transverse paths. It is obtained by integrating an infinitesimal measure which generalizes the norm on the tangent space. This requires the definition and the study of the metric tangent space (in Gromov's sense). As an example, we compute those measures in the case of contact sub-Riemannian manifolds.

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References

  1. A. Agrachev, B. Bonnard, M. Chyba and I. Kupka,Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var.2 (1997), 377–448.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Beals, B. Gaveau and P. C. Greiner,Hamilton-Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl.79 (2000), 633–689.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Bellaïche,The tangent space in sub-Riemannian geometry, inSub-Riemannian Geometry (A. Bellaïche and J.-J. Risler, eds.), Progress in Mathematics, Birkhäuser, Boston, 1996.

    Google Scholar 

  4. M. Gromov,Structures Métriques pour les Variétés Riemanniemes, Cedic-Nathan, Paris, 1981.

    Google Scholar 

  5. M. Gromov,Carnot-Carathéodory spaces seen from within, inSub-Riemannian Geometry (A. Bellaïche and J.-J. Risler, eds.), Progress in Mathematics, Birkhäuser, Boston, 1996.

    Google Scholar 

  6. H. Hermes,Nilpotent and high-order approximations of vector field systems, SIAM Review33 (1991), 238–264.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Jean,Paths in sub-Riemannian geometry, inNonlinear Control in the Year 2000 (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, eds.), Springer-Verlag, Berlin, 2000.

    Google Scholar 

  8. F. Jean,Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var.9 (2003), 485–506.

    MATH  MathSciNet  Google Scholar 

  9. F. Jean,Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems7 (2001), 473–500.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. A. Margulis and G. D. Mostow,The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal.5 (1995), 402–433.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. A. Margulis and G. D. Mostow,Some remarks on the definition of tangent cones in a Carnot-Carthéodory space, J. Analyse Math.80 (2000), 299–317.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Mitchell,On Carnot-Carathéodory metrics, J. Differential Geom.21 (1985), 35–45.

    MATH  MathSciNet  Google Scholar 

  13. R. Montgomery,Survey of singular geodesics, inSub-Riemannian Geometry (A. Bellaïche and J.-J. Risler, eds.), Progress in Mathematics, Birkhäuser, Boston, 1996.

    Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques, Université Paris 6, 175 rue du Chevaleret, 75013, Paris, France

    Elisha Falbel

  2. Laboratoire de Mathématiques Appliquées, école Nationale Supérieure des Techniques Avancées, 32, BD Victor, 75739, Paris, France

    Frédéric Jean

Authors
  1. Elisha Falbel
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  2. Frédéric Jean
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Falbel, E., Jean, F. Measures of transverse paths in sub-Riemannian geometry. J. Anal. Math. 91, 231–246 (2003). https://doi.org/10.1007/BF02788789

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  • DOI: https://doi.org/10.1007/BF02788789

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