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On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

Published online by Cambridge University Press:  17 April 2009

Kang-Hai Tan  and
Xiao-Ping Yang
Kang-Hai Tan
Affiliation:
Department of Applied Mathematics, Nanjing University of Science and Technology, 210094, Nanjing, Jiangsu, Peoples Republic of China e-mail: [email protected], [email protected]
Xiao-Ping Yang
Affiliation:
Department of Applied Mathematics, Nanjing University of Science and Technology, 210094, Nanjing, Jiangsu, Peoples Republic of China e-mail: [email protected], [email protected]
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We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 177 - 198
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bellaïche, A., ‘Tangent spaces in sub-Riemannian geometry’, in Sub-Riemannian Geometry, (Bellaïche, A. and Risler, J.J., Editors), Progress in Math. 144 (Birkhäuser, Basel, 1996), pp. 1–78.Google Scholar
[2]Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E. and Murray, R.M., ‘Nonholonomic mechanical systems with symmetry’, Arch. Rational Mech. Anal. 136 (1996), 21–99.Google Scholar
[3]Cartan, E., ‘Sur la representation geometrique des syst‘emes materials non holonomes’, Proc. Int. Congr. Math. 4 (1928), 253–261.Google Scholar
[4]Chow, W.L., ‘über Systeme non linearen partiellen Differentialgleichungen erster Ordung’, Math. Ann. 117 (1939), 98–105.Google Scholar
[5]Danielli, D., Garofalo, N. and Nhien, D.M., ‘Minimal surfaces, surfaces of constant mean curvature and isoperimetry in Carnot groups’, (preprint).Google Scholar
[6]Danielli, D., Garofalo, N. and Nhien, D.M., ‘Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces’, Mem. Amer. Math. Soc. (to appear).Google Scholar
[7]Derridj, M., ‘Un probléme aux limites pour une classe d'opérateurs du second ordre hypoelliptiques’, Ann. Inst. Fourier (Grenoble) 21 (1971), 99–148.Google Scholar
[8]Derridj, M., ‘Sur un théorème de traces’, Ann. Inst. Fourier (Grenoble) 22 (1972), 73–83.Google Scholar
[9]Franchi, B., Serapioni, R. and Cassano, F. Serra, ‘Rectifiability and perimeter in the Heisenberg group’, Math. Ann. 321 (2001), 479–531.Google Scholar
[10]Franchi, B., Serapioni, R. and Cassano, F. Serra, ‘Regular hypersurfaces, intrinsic perimeter and implicit functions theorem in Carnot groups’, Comm. Anal. Geom. (2003), 909–944.Google Scholar
[11]Franchi, B., Serapioni, R. and Cassano, F. Serra, ‘On the structure of finite perimeter sets in step 2 Carnot groups’, (preprint available from http://cvgmt.sns.it).Google Scholar
[12]Franchi, B., Serapioni, R. and Cassano, F. Serra, ‘Rectifiability and perimeter in step 2 groups’, (preprint available from http://cvgmt.sns.it).Google Scholar
[13]Garofalo, N. and Nhieu, D.M., ‘Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimimal surfaces’, Comm. Pure. Appl. Math. 49 (1996), 1081–1144.Google Scholar
[14]Garofalo, N. and Pauls, S.D., ‘The Bernstein problem in the Heisenberg group’, (preprint available from http://cvgmt.sns.it).Google Scholar
[15]Gorbatenko, E.M., ‘Differential geometry of nonholonomic manifolds (after V.V.Vagner)’, Ukrain. Geom. Sb. (1985), 31–43.Google Scholar
[16]Gromov, M., ‘Carnot-Carathéodory spaces seen from within’, in Sub-Riemannian Geometry, (by Bellaïche, A. and Risler, J.J., Editors), Progress in Math. 144 (Birkhäuser, Basel, 1996), pp. 79–323.CrossRefGoogle Scholar
[17]Koiller, J., Rodrigues, P.R. and Pitanga, P., ‘Non-holonomic connections following Élie Cartan’, An. Acad. Bras. Cienc. 7 (2001), 165–190.Google Scholar
[18]Koschorke, U., Vector fields and other vector bundle morphisms-a singularity approach, Lecture notes in Mathematics 847 (Springer-Verlag, Berlin, 1981).Google Scholar
[19]Magnani, V., Elements of geometric measure theory on sub-Riemannian groups, Ph.D. thesis, Scuola Normale Superiore, Pisa, 2002.Google Scholar
[20]Magnani, V., ‘A blow-up theorem for regular hypersurfaces on nilpotent groups’, Manuscripta Math. 110 (2003), 55–76.CrossRefGoogle Scholar
[21]Montgomery, R., A tour of Subriemannian geometry, their geodesics and applications, Mathematical Surveys and Monographs 91 (American Mathematical Society, Providence, R.I., 2002).Google Scholar
[22]Pauls, S.D., ‘Nonparametric minimal surfaces in the Heisenberg group’, Geometriae Dedicata (to appear).Google Scholar
[23]Pauls, S.D., ‘A notion of rectifiability modelled on Carnot groups’, Indiana Univ. Math. J. 53 (2004), 49–81.Google Scholar
[24]Schouten, J.A., ‘On nonholonomic connections’, Proc. Amsterdam Nederl. Akad. Wetensch. Ser. A, 31 (1928), 291–299.Google Scholar
[25]Sussmann, H.J., ‘Orbits of families of vector fields and integrability of distributions’, Trans. Amer. Math. Soc. 80 (1973), 171–188.Google Scholar
[26]Tan, K.H. and Yang, X.P., ‘Horizontal connection and horizontal mean curvature in Carnot groups’, (preprint).Google Scholar
[27]Vagner, V.V., ‘Differential geometry of nonholonomic manifolds’, (in Russian), Tr. Semin. Vectorn. Tenzorn. Anal. 213 (1935), 269–314.Google Scholar
[28]Vagner, V.V., ‘Differential geometry of nonholonomic manifolds’, in The VIII-th International Competition for the N. I. Lobatschewski Prize Report (The Kazan Physico-Mathematical Society, Kazan, 1940).Google Scholar
[29]Vagner, V.V., ‘A geometric interpretation of nonholonomic dynamical systems’, Tr. Semin. Vectorn. Tenzorn. Anal. 5 (1941), 301–327.Google Scholar
[30]Vagner, V.V., Geometria del calcolo delle variazioni 2 (C.I.M.E., Roma).Google Scholar
[31]Vershik, A.M. and Faddeev, L.D., ‘Lagrange mechanics in an invariant form’, in Probl. Theor. Phys. Vol.II, pp. 129–141. English translation: Sel. Math. Sov. 1 (1981), 339–350.Google Scholar
[32]Vershik, A.M. and Gershkovich, V.Ya., ‘Nonholomic problems and the theory of distributions’, Acta. Appl. Math. 12 (1988), 181–209.CrossRefGoogle Scholar
[33]Vershik, A.M. and Gershkovich, V.Ya., ‘Nonholonomic dynamical systems, geometry of distributions and variational problems’, in Dynamical Systems VII, (Arnold, V.I. and Novikov, S.P., Editors), Encyclopaedia of Mathematics Sciences 16 (Springer-Verlag, 1994).Google Scholar