Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-07T17:17:25.921Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLASSIFICATION OF SPHERICAL SYMMETRIC CR MANIFOLDS

Part of: Global differential geometry

Published online by Cambridge University Press:  08 June 2009

G. DILEO  and
A. LOTTA
G. DILEO
Affiliation:
Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari, Italy (email: dileo@dm.uniba.it)
A. LOTTA*
Affiliation:
Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari, Italy (email: lotta@dm.uniba.it)
*
✉For correspondence; e-mail: lotta@dm.uniba.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we get different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka–Webster connection and of the Riemannian curvature tensor. As a consequence we obtain the classification of the simply connected, spherical symmetric pseudo-Hermitian manifolds.

Keywords

CR-symmetric spacespherical CR manifoldcontact Riemannian (k,μ)-spaceBochner curvature tensor

MSC classification

Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C35: Symmetric spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 251 - 274
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1] Abbassi, M. T. K., ‘Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M,g)’, Comment. Math. Univ. Carolin. 45(4) (2004), 591–596.Google Scholar
[2] Abbassi, M. T. K. and Sarih, M., ‘On Riemannian g-natural metrics of the form ag s+bg h+cg v on the tangent bundle of a Riemannian manifold (M,g)’, Mediterr. J. Math. 2(1) (2005), 19–43.CrossRefGoogle Scholar
[3] Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, vol. 203 (Birkhäuser, Boston, 2002).CrossRefGoogle Scholar
[4] Blair, D. E., Koufogiorgos, T. and Papantoniou, B. J., ‘Contact metric manifolds satisfying a nullity condition’, Israel J. Math. 91 (1995), 189–214.CrossRefGoogle Scholar
[5] Blair, D. E. and Vanhecke, L., ‘Symmetries and φ-symmetric spaces’, Tohoku Math. J. 39 (1987), 373–383.CrossRefGoogle Scholar
[6] Boeckx, E., ‘A class of locally φ-symmetric contact metric spaces’, Arch. Math. 72 (1999), 466–472.CrossRefGoogle Scholar
[7] Boeckx, E., ‘A full classification of contact metric (k,μ)-spaces’, Illinois J. Math. 44(1) (2000), 212–219.CrossRefGoogle Scholar
[8] Boeckx, E., ‘Contact-homogeneous locally φ-symmetric manifolds’, Glasg. Math. J. 48(1) (2006), 93–109.CrossRefGoogle Scholar
[9] Boeckx, E. and Cho, J. T., ‘η-parallel contact metric spaces’, Differential Geom. Appl. 22 (2005), 275–285.CrossRefGoogle Scholar
[10] Bryant, R. L., ‘Bochner-Kähler metrics’, J. Amer. Math. Soc. 14(3) (2001), 623–715 (electronic).CrossRefGoogle Scholar
[11] Burns, D. and Shnider, S., ‘Spherical hypersurfaces in complex manifolds’, Invent. Math. 33(3) (1976), 223–246.CrossRefGoogle Scholar
[12] Chern, S. S. and Moser, J. K., ‘Real hypersurfaces in complex manifolds’, Acta Math. 133 (1974), 219–271.CrossRefGoogle Scholar
[13] Cho, J. T., ‘Geometry of contact strongly pseudo-convex CR-manifolds’, J. Korean Math. Soc. 43(5) (2006), 1019–1045.CrossRefGoogle Scholar
[14] David, L., ‘Weyl connections and curvature properties of CR manifolds’, Ann. Global Anal. Geom. 26(1) (2004), 59–72.CrossRefGoogle Scholar
[15] Ghosh, A. and Sharma, R., ‘On contact strongly pseudo-convex integrable CR manifolds.’, J. Geom. 66(1–2) (1999), 116–122.CrossRefGoogle Scholar
[16] Falcitelli, M., Ianus, S. and Pastore, A. M., Riemannian Submersions and Related Topics (World Scientific, River Edge, NJ, 2004).Google Scholar
[17] Feldmueller, D. and Lehmann, R., ‘Homogeneous CR-hypersurface-structures on spheres’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14(4) (1987), 513–525.Google Scholar
[18] Jiménez, J. A. and Kowalski, O., ‘The classification of φ-symmetric Sasakian manifolds’, Monatsh. Math. 115 (1993), 83–98.CrossRefGoogle Scholar
[19] Kaup, W. and Zaitsev, D., ‘On symmetric Cauchy–Riemann manifolds’, Adv. Math. 149 (2000), 145–181.CrossRefGoogle Scholar
[20] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. I (Interscience Publishers, New York, 1963).Google Scholar
[21] Kowalski, O. and Sekizawa, M., ‘On tangent sphere bundles with small or large constant radius’, Ann. Global Anal. Geom. 18(3–4) (2000), 207–219.CrossRefGoogle Scholar
[22] Matsumoto, M. and Tanno, S., ‘Kählerian spaces with parallel or vanishing Bochner curvature tensor’, Tensor (N.S.) 27 (1973), 291–294.Google Scholar
[23] Sakamoto, K. and Takemura, Y., ‘On almost contact structures belonging to a CR-structure’, Kodai Math. J. 3 (1980), 144–161.CrossRefGoogle Scholar
[24] Sakamoto, K. and Takemura, Y., ‘Curvature invariants of CR-manifolds’, Kodai Math. J. 4 (1981), 251–265.CrossRefGoogle Scholar
[25] Takahashi, T., ‘Sasakian φ-symmetric spaces’, Tohoku Math. J. 29 (1977), 91–113.CrossRefGoogle Scholar
[26] Tanaka, N., ‘On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections’, Japan. J. Math. 20 (1976), 131–190.CrossRefGoogle Scholar
[27] Tanno, S., ‘Variational problems on contact Riemannian manifolds’, Trans. Amer. Math. Soc. 314 (1989), 349–379.CrossRefGoogle Scholar
[28] Tanno, S., ‘The standard CR structure on the unit tangent bundle’, Tohoku Math. J. 44(4) (1992), 535–543.CrossRefGoogle Scholar
[29] Tanno, S., ‘Pseudo-conformal invariants of type (1,3) of CR manifolds’, Hokkaido Math. J. 20(2) (1991), 195–204.CrossRefGoogle Scholar
[30] Tashiro, Y., ‘On contact structures of tangent sphere bundles’, Tohoku Math. J. 21 (1969), 117–143.CrossRefGoogle Scholar
[31] Webster, S. M., ‘Pseudo-Hermitian structures on a real hypersurface’, J. Differential Geom. 13(1) (1978), 25–41.CrossRefGoogle Scholar
[32] Webster, S. M., ‘On the pseudo-conformal geometry of a Kähler manifold’, Math. Z. 157 (1977), 265–270.CrossRefGoogle Scholar