The local geometry of Carnot manifolds at singular points

S. V. Selivanova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2011), pp. 94-97.

Résumé

In this paper we study the local geometry of Carnot manifolds in a neighborhood of a singular point in the case when horizontal vector fields are $2M$-smooth. Here $M$ is the depth of a Carnot manifold.

Détail

Keywords: Carnot manifold, singular point
Mots-clés : tangent cone.

Bibliographie

[1] Gromov M., “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian geometry. Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323 | MR | Zbl

[2] Mitchell J., “On Carnot–Caratheodory metrics”, J. Different. Geom., 21 (1985), 35–45 | MR | Zbl

[3] Karmanova M., Vodop'yanov S., “Geometry of Carnot–Carathéodory spaces, differentiability, coarea and area formulas”, Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser, Basel, 2009, 233–335 | MR

[4] Vodopyanov S. K., Karmanova M. B., “Lokalnaya approksimatsionnaya teorema na mnogoobraziyakh Karno v usloviyakh minimalnoi gladkosti”, Dokl. RAN, 413:3 (2007), 305–311 | MR

[5] Greshnov A. V., “O prilozhenii metodov gruppovogo analiza differentsialnykh uravnenii k nekotorym sistemam $C^1$-gladkikh vektornykh polei”, Sib. matem. zhurn., 50:1 (2009), 47–62 | MR | Zbl

[6] Karmanova M. B., “Novyi podkhod k issledovaniyu prostranstv Karno–Karateodori”, Dokl. RAN, 434:3 (2010), 309–314 | MR

[7] Selivanova S. V., “Kasatelnyi konus k regulyarnomu kvazimetricheskomu prostranstvu Karno–Karateodori”, Dokl. RAN, 425:5 (2009), 595–599 | MR | Zbl

[8] Selivanova S. V., “Kasatelnyi konus k kvazimetricheskomu prostranstvu s rastyazheniyami”, Sib. matem. zhurn., 51:2 (2010), 388–403 | MR | Zbl

[9] Bellaïche A., “The tangent space in sub-Riemannian geometry”, Sub-Riemannian Geometry. Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | MR | Zbl

[10] Hermes H., “Nilpotent and high-order approximations of vector field systems”, SIAM Rev., 33:2 (1991), 238–264 | DOI | MR | Zbl

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

[12] Jean F., “Uniform estimation of sub-Riemannian balls”, J. Dyn. and Control Syst., 7:4 (2001), 473–500 | DOI | MR | Zbl

[13] Agrachev A. A., Sachkov Yu. L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005

[14] Hörmander L., “Hypoelliptic second order differential equations”, Acta Math., 119:3–4 (1967), 147–171 | DOI | MR

[15] Stein E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl

[16] Christ M., Nagel A., Stein E. M., Wainger S., “Singular and maximal Radon transforms: analysis and geometry”, Ann. Math. Ser. 2, 150:2 (1999), 489–577 | DOI | MR | Zbl

[17] Rashevskii P. K., “O soedinimosti lyubykh dvukh tochek vpolne negolonomnogo prostranstva dopustimoi liniei”, Uchen. zap. Mosk. gos. ped. in-ta im. K. Libknekhta. Ser. fiz.-matem., 3:2 (1938), 83–94

[18] Chow W. L., “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung”, Math. Ann., 117 (1939), 98–105 | DOI | MR | Zbl

[19] Gromov M., “Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits)”, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73 | DOI | MR

[20] Burago D. Yu., Burago Yu. D., Ivanov S. V., Kurs metricheskoi geometrii, In-t kompyuternykh issledovanii, M.–Izhevsk, 2004