On the generalized triangle inequality for quasimetrics induced by noncommuting vector fields

A. V. Greshnov

Matematičeskie trudy, Tome 14 (2011) no. 1, pp. 70-98.

Résumé

For a sufficiently wide class of $r$-smooth basis vector fields, we obtain necessary and sufficient conditions for some anisotropic metric functions induced by these vector fields to be quasimetrics. These results are applied to the problem of the existence of a nilpotent tangent cone at a distinguished point.

Bibliographie

[1] Akivis M. A., “O kanonicheskikh razlozheniyakh uravnenii lokalnoi analiticheskoi kvazigruppy”, Dokl. AN SSSR, 188 (1969), 967–970 | MR | Zbl

[2] Vodopyanov S. K., Greshnov A. V., “O differentsiruemosti otobrazhenii prostranstv Karno–Karateodori”, Dokl. RAN, 389:5 (2003), 592–596 | MR

[3] Vodopyanov S. K., Karmanova M. B., “Subrimanova geometriya pri minimalnoi gladkosti vektornykh polei”, Dokl. RAN, 422:5 (2008), 583–588 | MR

[4] Greshnov A. V., “Metriki ravnomerno regulyarnykh prostranstv Karno–Karateodori i ikh kasatelnykh konusov”, Sib. matem. zhurn., 47:2 (2006), 259–292 | MR | Zbl

[5] Greshnov A. V., “Lokalnaya approksimatsiya ravnomerno regulyarnykh kvaziprostranstv Karno–Karateodori ikh kasatelnymi konusami”, Sib. matem. zhurn., 48:2 (2007), 290–312 | MR | Zbl

[6] Greshnov A. V., “O differentsiruemosti gorizontalnykh krivykh v kvaziprostranstvakh Karno–Karateodori”, Sib. matem. zhurn., 49:1 (2008), 67–86 | MR | Zbl

[7] Greshnov A. V., “O primenenii metodov gruppovogo analiza differentsialnykh uravnenii dlya nekotorykh sistem $C^1$-gladkikh vektornykh polei”, Sib. matem. zhurn., 50:1 (2009), 47–62 | MR | Zbl

[8] Greshnov A. V., “O primenenii formuly Teilora na nekotorykh kvaziprostranstvakh”, Matem. tr., 12:1 (2009), 3–25 | MR

[9] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR | Zbl

[10] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Fizmatgiz, M., 1961 | MR

[11] Belläiche A., “The tangent space in sub-Riemannian geometry”, Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, 1–78 | MR | Zbl

[12] Bonfiglioli A., Lanconelli E., Uguzzoni F., Stratified Lie Groups and Potential Theory for their Sub\@Laplacians, Springer Monographs in Mathematics, Springer-Verlag, Berlin–Heidelberg, 2007 | MR | Zbl

[13] Goodman R., “Lifting vector fields to nilpotent Lie groups”, J. Math. Pures Appl. (9), 57:1 (1978), 77–86 | MR

[14] Gromov M., “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, 79–323 | MR | Zbl

[15] Koranyi A., Reimann H. M., “Foundations for the theory of quasiconformal mappings on the Heisenberg group”, Adv. Math., 111:1 (1995), 1–87 | DOI | MR | Zbl

[16] Margulis G. A., Mostow G. D., “The differential of quasi-conformal mapping of a Carnot–Carathéodory space”, Geom. Funct. Anal., 5:2 (1995), 402–433 | DOI | MR | Zbl

[17] Métivier G., “Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques”, Comm. Partial Differential Equations, 1:5 (1976), 467–519 | DOI | MR | Zbl

[18] Mitchell J., “On Carnot–Carathéodory metrics”, J. Differential Geom., 21:1 (1985), 35–45 | MR | Zbl

[19] Nagel A., Stein E. M., Wainger S., “Balls and metrics defined by vector fields. I: Basic properties”, Acta Math., 155:1–2 (1985), 103–147 | DOI | MR | Zbl

[20] Rothchild L. P., Stein E. M., “Hypoelliptic differential operators and nilpotent groups”, Acta Math., 137:3–4 (1976), 247–320 | DOI | MR

[21] Stein E. M., Harmonic Analysis: Real-Variables Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43{, Monographs in Harmonic Analysis,} III, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl