Géométrie sous-riemannienne

Ivan Kupka

Séminaire Bourbaki (1995-1996)

  • Volume: 38, page 351-380
  • ISSN: 0303-1179

How to cite

top

Kupka, Ivan. "Géométrie sous-riemannienne." Séminaire Bourbaki 38 (1995-1996): 351-380. <http://eudml.org/doc/110220>.

@article{Kupka1995-1996,
author = {Kupka, Ivan},
journal = {Séminaire Bourbaki},
keywords = {Carnot-Carathéodory metric; isoperimetric inequality; sub-Riemannian manifolds; geodesics; Hausdorff dimension; hypoelliptic operators},
language = {fre},
pages = {351-380},
publisher = {Société Mathématique de France},
title = {Géométrie sous-riemannienne},
url = {http://eudml.org/doc/110220},
volume = {38},
year = {1995-1996},
}

TY - JOUR
AU - Kupka, Ivan
TI - Géométrie sous-riemannienne
JO - Séminaire Bourbaki
PY - 1995-1996
PB - Société Mathématique de France
VL - 38
SP - 351
EP - 380
LA - fre
KW - Carnot-Carathéodory metric; isoperimetric inequality; sub-Riemannian manifolds; geodesics; Hausdorff dimension; hypoelliptic operators
UR - http://eudml.org/doc/110220
ER -

References

top
  1. [Ag] A.A. Agrachev - Methods of control theory in non-holonomic geometry, Proceedings ICM-94, Zürich, Birkhäuser. Zbl0848.93012
  2. [Ag1] A.A. Agrachev - Any smooth simple H1-local length minimizer in the Carnot-Caratheodory space is a C0-local length minimizer, preprint, Dijon, juin 1996. 
  3. [A-S1] A.A. Agrachev et A.V. Sarychev - Strong minimality of abnormal geodesics for 2-distributions, Journal of Dynamical Systems and Control Systems1, n° 2 (1995), 139-176. Zbl0951.53029MR1333769
  4. [A-S2] A.A. Agrachev et A.V. Sarychev - Abnormal sub-riemannian geodesics : Morse index and rigidity, à paraître dans les Annales I.H.P. Zbl0866.58023
  5. [B-G-G1] R. Beals, B. Gaveau et P. Greiner - The Green function for a model twostep hypoelliptic operator and the analysis of certain tangential Cauchy-Riemann complexes, preprint (1995). 
  6. [B-G-G2] R. Beals, B. Gaveau et P. Greiner - Subelliptic geometry, preprint (1995). MR1486587
  7. [B-G-G3] R. Beals, B. Gaveau et P. Greiner - On a geometric formula for the fundamental solution of subelliptic laplacians, preprint (1995). MR1409074
  8. [B-G-G4] R. Beals, B. Gaveau et P. Greiner - Complex Hamiltonian mechanisms and parametrics for subelliptic laplacians, preprint (1995). 
  9. [Bell] A. Bellaïche - The tangent space in sub-riemannian geometry, dans "Journées non holonomes", A. Bellaïche-J.-J. Risler eds, Birkhäuser, Progress in Math., vol. 144 (1996), 1-78. Zbl0862.53031MR1421822
  10. [Ben] G. Ben Arous - Développement du noyau de la chaleur sur la diagonale, Annales Inst. Four.39 (1) (1989), 73-99. Zbl0659.35024MR1011978
  11. [B-L] G. Ben Arous et R. Léandre - Décroissance exponentielle du noyau de la chaleur sur la diagonale, Prob. Theory Relat. Fields. Part. I : 90 (1991), 175-202 ; Part. II : 90 (1991), 377-402. Zbl0734.60027
  12. [Bi] J.-M. Bismut - Large deviations and the Malliavin calculus, Birkhäuser, Progress in Math., vol. 45 (1984). Zbl0537.35003MR755001
  13. [B-H] R. Bryant et L. Hsu - Rigidity of integral curves of rank 2 distributions, Invent. Math.114 (1993), 435-461. Zbl0807.58007MR1240644
  14. [B-Z] Yu.D. Burago et V.A. Zagaller - Geometric inequalities, Springer GdMW Bd 285 (1988). Zbl0633.53002MR936419
  15. [C-K-S] E.A. Carlen, S. Kusuoka et D.W. Stroock - Upper bounds for symmetric Markov transition functions, Ann. Inst. Henri Poincaré, Probabilité et Statistique, supp. au n° 2 (1987), 245-287. Zbl0634.60066MR898496
  16. [E-G-K] El-Houcine, J.-P. Gauthier et I. Kupka - Small sub-riemannian balls in R3, Journal of Dynamical and Control Systems, vol. 2, n° 3, July 1996, 359-421. Zbl0941.53024MR1403263
  17. [Fa-G] E. Falbel et C. Gorodski - On contact symmetric sub-riemannian spaces, Annales Scient. E.N.S.28 (1995), 571-589. Zbl0836.53032MR1341662
  18. [Fa-G-R] E. Falbel, C. Gorodski et M. Rumin - On the holonomy of sub-riemannian spaces, preprint (1995). 
  19. [F-P] C.L. Feffermann et D.H. Phong - Subelliptic eigenvalue problems, dans Proceedings Conference on Harmonie analysis in honor of Antony Zygmund, Math. Ser., Wadsworth, Belmont CA, 590-606. Zbl0503.35071MR730094
  20. [F-S] C.L. Feffermann et A. Sanchez-Calle - Fundamental solutions of second order subelliptic operators, Ann. of Math.124 (1986), 247-272. Zbl0613.35002MR855295
  21. [Fl-L] P. Florchinger et R. Léandre - Décroissance non-exponentielle du noyau de la chaleur, Prob. Theory Relat. Fields95 (1993), 237-262. Zbl0791.60039MR1214089
  22. [Ga1] B. Gaveau - Principe de moindre action ; propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math.139 (1977), 95-153. Zbl0366.22010MR461589
  23. [Ga2] B. Gaveau - Systèmes dynamiques associés à certains opérateurs hypoelliptiques, Bulletin Sci. Math., 2ème série, 102 (1978), 203-229. Zbl0391.35019MR509102
  24. [Go1] R. Goodman - Nilpotent Lie groups : structure and applications to analysis, Springer LNM562 (1976). Zbl0347.22001MR442149
  25. [Go2] R. Goodman - Lifting vector fields to nilpotent Lie groups, Journ. Math. Pures et Appl.57 (1978), 77-86. Zbl0336.57015MR494316
  26. [Gr1] M. Gromov - Carnot-Caratheodory spaces seen from within, dans le même volume que [Bell], 79-323. MR1421823
  27. [Gr2] M. Gromov - Partial differential relations, Springer Ergebnisse der Mathematik und ihrer Grenzgebiete3, Folge, Band 9 (1986). Zbl0651.53001MR864505
  28. [G-L-P] M. Gromov, J. Lafontaine et P. Pansu - Structures métriques pour les variétés riemanniennes, Cedic-Fernand Nathan (1981). Zbl0509.53034MR682063
  29. [Ha] U. Hamenstädt - Some regularity theorems for Carnot-Caratheodory metrics, J. Diff. Geom.32 (1991), 192-201. Zbl0687.53041MR1078163
  30. [H-Le] A. Hilbert et R. Léandre - Nagel, Stein, Wainger estimates for balls associated with the Bismut condition, preprint. Zbl1006.60047
  31. [Hö1] L. Hörmander - Hypoelliptic second order differential equations, Acta Math.119 (1967), 147-171. Zbl0156.10701MR222474
  32. [Hö2] L. Hörmander - Analysis of linear partial differential operators, III, Springer GdMW274 (1985). Zbl0601.35001MR781536
  33. [J-S1] D. Jerison et A. Sanchez-Calle - Subelliptic second order differential operators, in Complex analysis III, Springer LNM1277 (1987). Zbl0634.35017MR922334
  34. [J-S2] D. Jerison et A. Sanchez-Calle - Estimates for the heat kernel for the sum of squares of vector fields, Indiana Univ. Math. Journ.35 (1986), 835-854. Zbl0639.58026MR865430
  35. [Le1] R. Léandre - Volumes des boules sous-riemanniennes et explosion du noyau de la chaleur au sens de Stein, Séminaire de Probabilité XXIII, J. Azéma, J. Yor eds, Springer LNM1372 (1990), 426-448. Zbl0764.60053MR1022930
  36. [Le2] R. Léandre - Développement asymptotique de la densité d'une diffusion dégénérée, Forum Math.4 (1992), 45-75. Zbl0749.60054MR1142473
  37. [Le3] R. Léandre - Applications quantitatives et géométriques du calcul de Malliavin, Stochastic Analysis, G. Métivier, S. Watanabe eds, Springer LNM1322 (1988), 109-133. Zbl0666.60014MR962867
  38. [L-S] W.S. Liu et H.J. Sussmann - Shortest paths for sub-riemannian metrics on rank 2 distributions, Mem. A.M.S., n° 564, Vol. 118, november 1995. Zbl0843.53038
  39. [M] P. Malliavin - Stochastic calculus of variations and hypoelliptic operators, Proc. Intern. Symp. S.D.E. Kyoto, 1976, K. Itô edt., 195-263, Kinokuniya, Tokyo, 1978. Zbl0411.60060MR536013
  40. [Ma] P. Mattila - Geometry of sets and measures in euclidean spaces : fractals and rectifiability, Cambridge studies in advanced mathematics, vol. 44, Cambridge University Press (1995). Zbl0819.28004MR1333890
  41. [Me] G. Métivier - Fonction spectrale et valeurs propres d'une classe d'opérateurs non-elliptiques, Comm. Partial Differential Equations1 (1976), 467-519. Zbl0376.35012MR427858
  42. [Mi] J. Mitchell - On Carnot-Caratheodory metrics, Jour. Diff. Geom.21 (1985), 35-45. Zbl0554.53023MR806700
  43. [Mo] S.A. Molchanov - Diffusion processes and riemannian geometry, Russian Math. Surveys30 (1975), 1-63. Zbl0315.53026MR413289
  44. [Mon] R. Montgomery - A survey of singular curves in sub-riemannian geometry, Journal of Dynamical and Control Systems, vol. 1, n° 1 (1995), 49-90. Zbl0941.53021MR1319057
  45. [Mu] S. Mustapha - Distances sous-elliptiques et puissances d'opérateurs diffé- rentiels, Ark. för Math.34 (1996), 159-177. Zbl0953.35037MR1396629
  46. [M-V] S. Mustapha et N. Varopoulos - Comparaison höldérienne des distances sous-elliptiques et calcul S(m, g), Potential Analysis4 (1995), 415-428. Zbl0840.35129MR1354893
  47. [N-S-W] A. Nagel, E.M. Stein et S. Wainger - Balls and metrics defined by vector fields I : basic properties, Acta Math.155 (1985), 103-147. Zbl0578.32044MR793239
  48. [O-R] O.A. Oleinik et E.V. Radkevic - Second order equations with non-negative characteristic form, A.M.S. (1973). 
  49. [P] P. Pansu - Métriques de Carnot-Caratheodory et quasi-isométries des espaces symétriques de rang un, Ann. of Math.129:1 (1989), 1-61. Zbl0678.53042MR979599
  50. [R] M. Rumin - Un complexe de formes différentielles sur les variétés de contact, C.R.A.S. t. 310 (1990), 401-404. Zbl0694.57010MR1046521
  51. [S] A. Sanchez-Calle - Fundamental solutions and geometry of the sums of squares of vector fields, Invent. Math.78 (1984), 143-160. Zbl0582.58004MR762360
  52. [Ste] E.M. Stein - Singular integrals and differentiability properties of functions, Princeton Univ. Press (1970). Zbl0207.13501MR290095
  53. [Str] R.S. Strichartz - Sub-riemannian geometry, Jour. Diff. Geometry24 (1986), 221-263. Zbl0609.53021MR862049
  54. [Th] R. Thom - Remarques sur les problèmes comportant des inégalités différentielles globales, Bull. Soc. Math. France87 (1959), 455-461. Zbl0213.25302MR121807
  55. [Va1] N.Th. Varopoulos - Small time Gaussian estimates of heat diffusion kernels, II : the theory of large derivations, Jour. Funct. Analysis93 (1990), 1-33. Zbl0712.58056MR1070036
  56. [Va2] N.Th. Varopoulos - Opérateurs sous-elliptiques du second ordre, C.R.A.S., t. 308, Série I (1989), 437-440. Zbl0699.35054MR991877
  57. [V-S-C] N.Th. Varopoulos, L. Saloff-Coste et Th. Coulhon - Analysis and geometry on groups, Cambridge Univ. Press (1993). MR1218884
  58. [V-G] A.M. Vershik et V.Ya. Gershkovich - Non-holonomic problems and the theory of distributions, Acta Appl. Math.12 (1988), 181-209. Zbl0666.58004MR966452

Citations in EuDML Documents

top
  1. A. Agrachev, B. Bonnard, M. Chyba, I. Kupka, Sub-riemannian sphere in Martinet flat case
  2. Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez, On complexity and motion planning for co-rank one sub-Riemannian metrics
  3. Frédéric Jean, Entropy and complexity of a path in sub-Riemannian geometry
  4. Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez, On complexity and motion planning for co-rank one sub-riemannian metrics
  5. Frédéric Jean, Entropy and complexity of a path in sub-riemannian geometry
  6. P. Cannarsa, L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls
  7. Bernard Bonnard, Monique Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
  8. Valentino Magnani, Blow-up of regular submanifolds in Heisenberg groups and applications
  9. Bernard Bonnard, Monique Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
  10. B. Bonnard, E. Trélat, On the role of abnormal minimizers in sub-riemannian geometry

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.