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Characterisation of the sub-Riemannian isometry groups of the H-type groups

Published online by Cambridge University Press:  17 April 2009

Kang-Hai Tan  and
Xiao-Ping Yang
Kang-Hai Tan
Affiliation:
Department of Applied Mathematics, Science School Nanjing University of Science and Technology, 210094, Nanjing, Peoples Republic China, e-mail: tankanghai2000@yahoo.com.cn, yangxp@mail.njust.edu.cn
Xiao-Ping Yang
Affiliation:
Department of Applied Mathematics, Science School Nanjing University of Science and Technology, 210094, Nanjing, Peoples Republic China, e-mail: tankanghai2000@yahoo.com.cn, yangxp@mail.njust.edu.cn
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For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesics. We use properties of sub-Riemannian geodesics in G to characterise the isometry group ISO(G) with respect to the Carnot-Carathéodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 87 - 100
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Aebischer, B., Borer, M., Kälin, M., Leuenberger, C. and Reimann, H.M., Symplectic geometry, Progress in Mathematics 124 (Birkhäuser, Basel, 1992).Google Scholar
[2]Ambrosio, L. and Rigot, S., ‘Optimal mass transportation in the Heisenberg group’, J. Funct. Anal. (to appear).Google Scholar
[3]Beals, R., Gaveau, B. and Greiner, P.C., ‘Hamilton-Jacobi theory and heat kernel on Heisenberg groups’, J. Math. Pures Appl. 79 (2000), 633–689.CrossRefGoogle Scholar
[4]Bellaíche, A., The tangent spaces in sub-Riemannian geometry, Progress in Mathematics 144 (Birkhäuser, Basel, 1996).CrossRefGoogle Scholar
[5]Berndt, J., Tricerri, F. and Vanhecke, L., Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math. 1598 (Springer-Verlag, Berlin, Heidelberg, New York 1994).Google Scholar
[6]Chow, W.L., ‘über Systeme non linearen partiellen Differentialgleichungen erster Ordung’, Math. Ann. 117 (1939), 98–105.CrossRefGoogle Scholar
[7]Folland, G.B. and Stein, E.M., Hardy spaces on homogeneous groups (Princeton Univ. Press, Princeton, N.J., 1982).Google Scholar
[8]Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152 (Birkhäuser, Boston, 1999).Google Scholar
[9]Gromov, M., Carnot-Carathéodory spaces seen from within, Progress in Mathematics 144 (Birkhäuser, Basel, 1996).Google Scholar
[10]Hutchinson, J.E., ‘Fractals and self similarity’, Indiana. Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[11]Kaplan, A., ‘Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms’, Trans. Amer. Math. Soc. 258 (1980), 147–153.CrossRefGoogle Scholar
[12]Kaplan, A., ‘Riemannian nilmanifolds attached to Clifford modules’, Geom. Dedicata, 11 (1981), 127–136.CrossRefGoogle Scholar
[13]Korányi, A., ‘Geometric properties of Heisenberg-type groups’, Adv. in Math. 56 (1985), 28–38.CrossRefGoogle Scholar
[14]Korányi, A. and Reimann, H.M., ‘Quasiconformal mappings on the Heisenberg group’, Invent. Math. 80 (1985), 309–338.CrossRefGoogle Scholar
[15]Koranyi, A., Geometric aspects of analysis on the Heisenberg group, Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982), 209C258 (Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983).Google Scholar
[16]Korányi, A. and Reimann, H.M., ‘Foundations for the theory of quasiconformal mappings on the Heisenberg group’, Adv. Math. 111 (1985), 1–87.CrossRefGoogle Scholar
[17]Leonardi, G.P. and Masnou, S., ‘On the isoperimetric problems in the Heisenberg group ℍn’, (preprint 2002) available from http://cvgmt.sns.it.Google Scholar
[18]Leonardi, G.P. and Rigot, S., ‘Isoperimetric sets on Carnot groups’, Houston J. Math. 29 (2003), 609–637.Google Scholar
[19]Liu, W.S. and Sussman, H., ‘Shortest paths for sub-Riemannian metrics on rank-two distributions’, Mem. Amer. Math. Soc. 118 (1995).Google Scholar
[20]Monti, R., ‘Some properties of Carnot-Caratheodory balls in the Heisenberg group’ Atti. Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Ren. Lincei (9) Mat. Appl. 11 (2000), 155–167.Google Scholar
[21]Montgomery, R., A tour of sub-Riemannian geometry, their geodesics and applications, Mathematical Surveys and Monographs 91 (American Mathematical Society, Providence R.I., 2002).Google Scholar
[22]Pansu, P., ‘Métriques de CC et quasiisométries des espaces symétriques de rang un’, Ann. of Math. 119 (1989), 1–60.CrossRefGoogle Scholar
[23]Stein, E.M., Harmonic analysis (Princeton University press, Princeton N.J., 1993).Google Scholar
[24]Tan, K.H., ‘Sobolev classes and horizontal energy minimizers between Carnot-Carathéodory spaces’, Commun. Contemp. Math. 6 (2004), 1–33.CrossRefGoogle Scholar