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Carnot-Carathéodory spaces seen from within

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Sub-Riemannian Geometry

Part of the book series: Progress in Mathematics ((PM,volume 144))

Abstract

Let V be a smooth manifold where we distinguish a subset H in the set of all piecewise smooth curves c in V. We assume that H is defined by a local condition on curves, i.e. if c is divided into segments c 1,…, c k , then

$$ c\, \in \,\mathcal{H} \Leftrightarrow \,{c_i} \in \mathcal{H},\,i = 1, \ldots ,k. $$

I thank Richard Montgomery for reading the manuscript and locating a multitude of errors.

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References

  1. Bass H., The degree of polynomial growth of finitely generated nilpotent group. Proc. Lond. Math. Soc. (3) 25, pp. 603–614, 1972.

    MathSciNet  MATH  Google Scholar 

  2. Bellaïche A., The tangent space in sub-Riemannian geometry. This volume.

    Google Scholar 

  3. Berestovskii V.N., Vershik A.M., Manifolds with intrinsic metrics and nonholonomic spaces. Advances in Sov. Math, vol (7) AMS 1992.

    Google Scholar 

  4. Bethuel F., The approximation problem for Sobolev maps betwen two manifolds. Acta. Math., 167 (1991), pp. 153–206.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bennequin D., Entrelacements et équations de Pfaff. Soc. Math. France, Astérisque 107–108 (1983), 83–161.

    MathSciNet  Google Scholar 

  6. Valére Bouche L., The geodesies’ problem in Sub-Riemannian geometry. About the R. Montgomery’s example simplified by I. Kupka. Preprint du L.A.M.A. 92-06 (1992).

    Google Scholar 

  7. Bryant R.L. and Griffiths P.A., Characteristic Cohomology of Differential systems (I): General Theory. J.-Amer.-Math.-Soc. 8 (1995), no. 3, 507–596.

    MathSciNet  MATH  Google Scholar 

  8. Brockett R.W., Nonlinear Control Theory and Differential Geometry, Proc. of the Int. Congress of Mathematicians, Warszawa, (1983).

    Google Scholar 

  9. Burago Y.D. and Zalgaller V.A.: Geometric inequalities. Springer, Berlin, Heidelberg 1988. [Russian edn.: Nauka, Moscow, 1980 ].

    MATH  Google Scholar 

  10. Chow W.L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Annalen 117 (1939), pp. 98–105.

    Article  Google Scholar 

  11. Corlette K., Hausdorff dimensions of limit sets I, preprint (1989).

    Google Scholar 

  12. D’Ambra G. and Gromov M., Lectures on transformation groups: geometry and dynamics, Surveys in Differential Geometry (Supplement to the Journal of Differential geometry), 1 (1991), pp. 19–111.

    MathSciNet  Google Scholar 

  13. D’Ambra G., Nash C 1-embedding theorem for Carnot-Carathéodory metrics. Differential-Geom.-Appl. 5 (1995), no. 2, 105–119.

    Article  MathSciNet  MATH  Google Scholar 

  14. D’Ambra G., Construction of connection inducing maps between principal bundles. J. Diff. Geom. 26, (1987), pp. 67–79.

    MathSciNet  MATH  Google Scholar 

  15. D’Ambra G., Induced subbundles and Nash’s implicit function theorem. Differential-Geom.-Appl. 4 (1994), no. 1, 91–105.

    Article  MathSciNet  MATH  Google Scholar 

  16. David G., Semmes S., Lipschitz mappings between dimensions, Unpublished manuscript.

    Google Scholar 

  17. Dektjarev I.M., Problems of value distribution in dimension higher than one, Uspeki Mat. Nauk 25: 6 (1970), pp. 53–84.

    MathSciNet  Google Scholar 

  18. Eells J. and Lemaire L., Another report on harmonic maps. Bull. London. Math. Soc. 20 (1988) 385–524, pp. 388.

    Article  MathSciNet  Google Scholar 

  19. Eliashberg Ya., Classification of overtwisted contact structures on manifolds, Invent. Math. 98 (1989), pp. 623–637.

    Article  MathSciNet  MATH  Google Scholar 

  20. Eliashberg Ya., Contact 3-manifolds, twenty years since J. Matinet’s work. Ann. Inst. Fourier 42 (1992), pp. 165–192.

    Article  MathSciNet  MATH  Google Scholar 

  21. Eliashberg Ya., New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991), pp. 513–520.

    Article  MathSciNet  MATH  Google Scholar 

  22. Eliashberg Ya., Filling by holomorphic disks and its applications, London Math. Soc. Lect. Notes Ser. 151 (1991), pp. 45–67.

    MathSciNet  Google Scholar 

  23. Falconer K., Fractal Geometry. Mathematical Foundations and Applications. John Wiley and Sons. 1990.

    Google Scholar 

  24. Fefferman C.L. and Phong D.H., Subelliptic eigenvalue problems, Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth Math. Series, pp. 590–606 (1981).

    Google Scholar 

  25. Folland G.B., Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv for Mat. 13, pp. 161–207 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  26. Fukawa K., Collapsing of Riemannian manifolds and eigenvalues of Laplacian operators, Invent. Math. 87 (1987), pp. 517–547.

    Google Scholar 

  27. Gromov M. and Eliashberg J., Construction of nonsingular isoperimetric films, Trudy Steklov Inst. 116, (1971), pp. 18–33.

    Google Scholar 

  28. Gromov M., Lafontaine J., Pansu P., Structures métriques pour les variétés Riemanniennes, Cedic-Fernand Nathan, Paris (1981).

    MATH  Google Scholar 

  29. Ge Z., Betti numbers, characteristic classes and sub-Riemannian geometry, Illinois J. of Mathematics, 36 (1992), pp. 372–403.

    MATH  Google Scholar 

  30. Ge Z., Collapsing Riemannian metrics to sub-Riemannian metrics and Laplacians to sub-Laplacians, Canadian J. Math., 1993, pp. 537–552.

    Google Scholar 

  31. Ge Z., On the Global Geometry of Three-dimensional Sub-Riemannian Manifolds. II On Sub-Riemannian Metrics and SL 2 R-geometry. The Fields Institute for Research in Mathematical Sciences, Preprint 1994.

    Google Scholar 

  32. Ge Z., Horizontal paths space and Carnot-Carathéodory metrics, Pacific J. of Mathematics, 161 (1993), pp. 255–286.

    MATH  Google Scholar 

  33. Ge Z., On a variational problem and the spaces of horizontal paths, Pacific J. of Mathematics, 149 (1991), pp. 61–93.

    MATH  Google Scholar 

  34. Giroux E., Topologie de contact en dimension 3, Séminaire Bourbaki, 1992–93, n° 760.

    Google Scholar 

  35. Goodman R.W., Nilpotent Groups, the Structure, and the Application to Analysis, Lectures Notes in Math., vol. 562, Springer, 1977.

    Google Scholar 

  36. Gromov M., Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991 ), 1–295. London Math. Soc. Lecture Note Ser., 182. Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

  37. Gromov M., Asymptotic geometry of homogeneous spaces, Proceedings Rend. Sem. Mat. Univ. Politec. Torino (1983), Conférence on Homogeneous Spaces in Symp. Math., pp. 59–60.

    Google Scholar 

  38. Gromov M., Convex integration of differential relations, Izv. Akad. Nauk. S.S.S.R. 33, 2, (1973), pp. 329–343.

    MathSciNet  Google Scholar 

  39. Gromov M., Dimension, non-linear spectra and width, Springer-Verlag, Lecture Notes in Mathematics, 1317 (1988), pp. 132-1-85.

    Google Scholar 

  40. Gromov M., Foliated plateau problem, parts I, II, Geometric and Functional Analysis 1: 1 (1991), pp. 14–79; (1991), pp. 253–320.

    Article  MathSciNet  MATH  Google Scholar 

  41. Gromov M., Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), pp. 1–147.

    MathSciNet  MATH  Google Scholar 

  42. Gromov M., Groups of polynomial growth and expanding maps, Publications Mathématiques IHES 53 (1981), pp. 53–73.

    MathSciNet  MATH  Google Scholar 

  43. Gromov M., Homotopical effects of dilatation. Journal of Differential Geometry 13 (1978) 303–310.

    MathSciNet  MATH  Google Scholar 

  44. Gromov M., Hyperbolic groups, Essays in Group Theory, S. Gersten editor, MSRI Publications n° 8, Springer (1987), pp. 75–265.

    Google Scholar 

  45. Gromov M., Hyperbolic manifolds, groups and actions, in “Riemannian Surfaces and Related Topics”, Ann. Math. Studies 97 (1981), pp. 183–215.

    MathSciNet  Google Scholar 

  46. Gromov M., Metric invariants of Kähler manifolds, Differential geometry and topology (Alghero, 1992 ), 90–116. World Sci. Publishing, River Edge, NJ, 1993.

    Google Scholar 

  47. Gromov M., Partial differential relations, Springer-Verlag (1986).

    MATH  Google Scholar 

  48. Gromov M., Paul Levy’s isoperimetric inequality, Preprint IHES (1980).

    Google Scholar 

  49. Gromov M., Stability and pinching. Sessions on Topology and Geometry of Manifolds (Italian) (Bologna, 1990 ), 55–97. Univ. Stud. Bologna, Bologna, 1992.

    Google Scholar 

  50. Gromov M., Systoles and intersystolic inequalities. Preprint IHES (1993).

    Google Scholar 

  51. Gromov M., Width and related invariants of Riemannian manifolds, Astérisque 163–164 (1988), pp. 93–109.

    Google Scholar 

  52. Hart P., Ordinary differential equations. John Wiley and Sons. New York-London-Sydney. 1964.

    Google Scholar 

  53. Hamenstädt U., Some regularity theorems for Carnot-Carathéodory metrics, J. Diff. Geom. 32 (1991), pp. 192–201.

    Google Scholar 

  54. Hermann R., Differential Geometry and the Calculus of Variations, Math. Sci. Eng. 49, Academic Press, New-York, 1968.

    Google Scholar 

  55. Hofer H., Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Inv. Math. 114 (3), pp. 515–585. 1993.

    Article  MathSciNet  MATH  Google Scholar 

  56. Holopainen I., Positive solutions of quasilinear elliptic equations on riemannian manifolds. Proc. London Math. Soc. (3) 65 (1992), pp. 651–672.

    MathSciNet  MATH  Google Scholar 

  57. Holopainen I. and Rickman S., Quasiregular mappings of the Heisenberg group. Preprint (1991).

    Google Scholar 

  58. Hörmander L., Hypoelliptic second order differential equations, Acta Math. 119, pp. 147–171 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  59. Hsu L., Calclus of Variations via the Griffiths formalism, Journal of Differential Geometry 36 (1992), pp. 551–589.

    MathSciNet  MATH  Google Scholar 

  60. Hsu L., Gromov’s h-principle for strongly bracket-generating distributions (IAS preprint: 1993)

    Google Scholar 

  61. Hsu L., Sub-Riemannian Comparison Theorems for Contact Manifolds (IAS preprint: 1993).

    Google Scholar 

  62. Jerison D. and Sánchez-Calle A., Subelliptic, second order differential operators. Lect. Notes in Math. 1277, pp. 46–77, Springer-Verlag 1987.

    Google Scholar 

  63. Jost J., Equilibrium Maps between Metric Spaces. Ruhr-Universität Bochum, Preprint.

    Google Scholar 

  64. Karcher H., Riemannian center of mass and mollifier smoothing. Comm. Pure and Appl. Math. 30 (1977), 509–541.

    Article  MathSciNet  MATH  Google Scholar 

  65. Karidi R., Geometry of balls in nilpotent Lie groups. Duke-Math.-J. 74 (1994), no. 2, 301–317.

    Article  MathSciNet  MATH  Google Scholar 

  66. Korevaar N., Upper bounds for eigenvalues of conformai metrics. J. Differential Geometry 37 (1993), pp. 73–93.

    MathSciNet  MATH  Google Scholar 

  67. Koranyi A. and Reimann H.M., Foundations for the theory of quasiconformal mappings on the Heisenberg group. Preprint 1991.

    Google Scholar 

  68. Lalonde F., Homologie de Shih d’une submersion (homologies non singulières des variétés feuilletées). Supplément au Bulletin de la Société Mathématique de France. 1978, Tome 115, Fascicule 4.

    Google Scholar 

  69. Lee Y.I., The metric properties of Lagrangian surfaces. Dissertation of Stanford University for the degree of doctor of philosophy. 1992.

    Google Scholar 

  70. Margulis G., Discrete groups of motions of manifolds of non-positive curvature (Russian), ICM 1974. Translated in A.M.S. Transi. (2) 109 (1977), pp. 33–45.

    MathSciNet  Google Scholar 

  71. Mitchell J., A Local Study of Carnot-Carathéodory Metrics. Ph.D. Thesis. Stony Brook 1982.

    Google Scholar 

  72. Mitchell J., On Carnot-Carathéodory metrics, J. Differ. Geom. 21 (1985), pp. 35–45.

    MATH  Google Scholar 

  73. Montgomery R., Survey of singular geodesies, this volume.

    Google Scholar 

  74. Mostow G.D., Strong rigidity of symmetric spaces, Ann. Math. Studies 78, Princeton (1973).

    Google Scholar 

  75. Nagel A., Stein E.M. and Wainger S., Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155, pp. 103–147 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  76. Nagata J-I., Modern dimension theory, North-Holland, 1965.

    MATH  Google Scholar 

  77. Pansu P., Croissance des boules et des géodésiques fermées dans les nil-variétés, Ergod. Th. dynam. Syst. 3 (1983), pp. 415–445.

    MathSciNet  MATH  Google Scholar 

  78. Pansu P., Une inégalité isopérimétrique sur le groupe d’Heisenberg, C.R. Acad. Sci. Paris 295 (1982), pp. 127–131.

    MathSciNet  MATH  Google Scholar 

  79. Pansu P., Métriques de Carnot-Carathéodory et quasi-isométries des espaces symétriques de rang un, Annals of Maths. 129: 1 (1989), pp. 1–61.

    Article  MathSciNet  MATH  Google Scholar 

  80. Pansu P., Quasiconformal mappings and manifolds of negative curvature, in “Curvature and Top of Riemannian Manifolds” (Shiohama et al., eds), Lect. Notes in Math. 1201 (1986), pp. 212–230, Springer-Verlag.

    Chapter  Google Scholar 

  81. Pansu P., Géometrie du groupe d’Heisenberg. Thèse Univ. Paris V II, 1982.

    Google Scholar 

  82. Pelletier F., et Valére Bouche L., Le problème des géodésiques en géométriesous-riemannienne singulière-II, C.-R.-Acad.-Sci.-Paris-Ser.-I-Math. 317 (1993), no. 1, 71–76.

    Google Scholar 

  83. Pesin Ya. B, Dimension type characteristics for invariant sets of dynamical systems. Uspekhi Mat. Nauk 43: 4 (1988), pp. 95–128.

    MathSciNet  Google Scholar 

  84. Pittet C., Isoperimetric Inequalities in Nilpotent Groups. Preprint 1994.

    Google Scholar 

  85. Rashevski P.K., About connecting two points of complete nonholonomic space by admissible curve, (in Russian), Uch. Zapiski ped. inst. Libknexta, no. 2, pp. 83–94, (1938).

    Google Scholar 

  86. Ruh E.A., Almost Lie groups, Proc ICM-1986, Berkeley, pp. 561–564, AMC 1987.

    Google Scholar 

  87. Rumin M., Formes différentielles sur les variétés de contact. Thèse de Doctorat. Orsay 1992.

    Google Scholar 

  88. Rumin M., Un complexe de formes différentielles sur les variétés de contact, C.R. Acad. Sci. Paris, t. 310 (1990), serie I., pp. 401–404.

    Google Scholar 

  89. Sarychev A.A.V., On the homotopy type of the spaces of trajectories of nonholonomic dynamic systems, Dokl. Acad. Sci. USSR, v. 314 (6), 1990.

    Google Scholar 

  90. Schoen R. and Uhlenbeck K., A regularity theory for harmonic maps. J. Differential Geom., 17 (1982), pp. 307–335.

    MathSciNet  MATH  Google Scholar 

  91. Singer I.M., Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13, pp. 685–697, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  92. Stein E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press (1970).

    MATH  Google Scholar 

  93. Strichartz R.S., Sub-Riemannian Geometry, J. Differential Geometry 24 (1986), pp. 221–263.

    MathSciNet  MATH  Google Scholar 

  94. Sussmann H.J., Abnormal sub-Riemannian minimizers, this volume.

    Google Scholar 

  95. Thom R., Remarques sur les problèmes comportant des inégalités différentielles globales, Bull. Soc. Math. France 87 (1959), pp. 455–461.

    MathSciNet  MATH  Google Scholar 

  96. Uhlenbeck Karen K., Connections with L p Bounds on Curvature. Commun. Math. Physics. 83, pp. 31–42 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  97. Varopoulos N. Th., Small Time Gaussian Estimates of Heat Diffusion Kernels. II. The Theory of Large Deviations. J. of Functional Analysis, Vol 93, No 1. (1990).

    Google Scholar 

  98. Varopoulos N. Th., Saloff-Coste L., Coulhon Th., Analysis and Geometry on Groups, Cambridge University Press (1993).

    Book  Google Scholar 

  99. Vershik A.M., Gershkovich V. Ya., nonholonomic Manifolds and Nilpotent Analysis, J. Geom. and Phys. vol. 5, no. 3, (1988).

    Google Scholar 

  100. Vershik A.M., Gershkovich V. Ya., Nonholonomic Geometry and Nilpotent Analysis, J. Geom. and Phys. 5 3 (1989), pp. 407–452.

    MathSciNet  Google Scholar 

  101. Vinogradov A.M., Geometry of nonlinear differential equations, J. Soviet Math. 17 (1981), pp. 1624–1649.

    Article  MATH  Google Scholar 

  102. Vinogradov A.M., The C-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 (pp. 1–129 ), 1984.

    Article  MathSciNet  MATH  Google Scholar 

  103. Vinogradov A.M., Local symmetries and conservation laws, Acta Applicandae Mathematicae 2 (21–78), 1984.

    Google Scholar 

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  1. Mikhael Gromov
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  1. Departement de Mathématiques, Université Paris 7 — Denis Diderot, 2, place Jussieu, F-75251, Paris 5e, France

    André Bellaïche

  2. Université Paris VI — Pierre et Marie Curie, F-75252, Paris 5e, France

    Jean-Jacques Risler

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© 1996 Birkhäuser Verlag Basel/Switzerland

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Gromov, M. (1996). Carnot-Carathéodory spaces seen from within. In: Bellaïche, A., Risler, JJ. (eds) Sub-Riemannian Geometry. Progress in Mathematics, vol 144. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9210-0_2

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  • DOI: https://doi.org/10.1007/978-3-0348-9210-0_2

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