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Parabolic Geodesics in Sasakian 3-Manifolds

Published online by Cambridge University Press:  20 November 2018

Jong Taek Cho ,
Jun-ichi Inoguchi  and
Ji-Eun Lee
Jong Taek Cho
Affiliation:
Department of Mathematics, Chonnam National University, Gwangju, 500–757, Korea e-mail: jtcho@chonnam.ac.kr
Jun-ichi Inoguchi
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan e-mail: inoguchi@sci.kj.yamagata-u.ac.jp
Ji-Eun Lee
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea e-mail: jieunlee12@gmail.com
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Abstract

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We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.

Keywords

58E20parabolic geodesicspseudo-Hermitian geometrySasakian manifolds
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 396 - 410
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Berger, M., Les variétés Riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa 15(1961), 179–246.Google Scholar
[2] Belkhelfa, M., Dillen, F., and Inoguchi, J. I., Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces. In: PDE’s, Submanifolds and Affine Differential Geometry (Warsaw, 2000), Banach Center Publ., 57, Polish Acad. Sci.,Warsaw, 2002, pp. 67–87.Google Scholar
[3] Bianchi, L., Sugli sazi a tre dimensioni che ammettono un gruppo continuo di movimenti. Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Tereza, Tomo XI (1898), 267–352; translation in Gen. Relativity and Gravitation 33(2001), no. 12, 2171–2252.Google Scholar
[4] Blair, D. E., Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics, 509, Springer-Verlag, Berlin-New York, 1976.Google Scholar
[5] Blair, D. E., Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, 203, Birkhäuser Boston Inc., Boston, MA, 2002.Google Scholar
[6] Caddeo, R., Montaldo, S., Oniciuc, C., and Piu, P., The classification of biharmonic curves of Cartan-Vranceanu 3-dimensional spaces. In: Modern trends in geometry and topology, Cluj Univ. Press, Cluj-Napoca, 2006, pp. 121–131.Google Scholar
[7] Caddeo, R., Oniciuc, C., and Piu, P., Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group. Rend. Sem. Mat. Univ. Politec. Torino 62(2004), no. 3, 265–277.Google Scholar
[8] Cartan, E., Leçons sur la géométrie des espaces de Riemann. Second Ed., Gauthier-Villards, Paris, 1946.Google Scholar
[9] Cho, J. T., Inoguchi, J., and Lee, J. E., Slant curves in Sasakian 3-manifolds. Bull. Austral. Math. Soc. 74(2006), no. 3, 359–367. doi:10.1017/S0004972700040429Google Scholar
[10] Cho, J. T., Inoguchi, J., and Lee, J. E., Biharmonic curves in 3-dimensional Sasakian space forms. Ann. Mat. Pura Appl. 186(2007), no. 4, 685–701. doi:10.1007/s10231-006-0026-xGoogle Scholar
[11] Cho, J. T., Inoguchi, J., and Lee, J. E., Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry. Abh. Math. Semin. Univ. Hambg. 79(2009), no. 1, 113–133. doi:10.1007/s12188-008-0014-8tGoogle Scholar
[12] Cho, J. T. and Lee, J. E., Slant curves in contact pseudo-Hermitian 3-manifold. Bull. Aust. Math. Soc. 78(2008), no. 3, 383–396. doi:10.1017/S0004972708000737Google Scholar
[13] Folland, G. B. and Stein, E. M., Estimates for the -complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27(1974), 429–522. doi:10.1002/cpa.3160270403Google Scholar
[14] Jerison, D. and Lee, J. M., Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differential Geom. 29(1989), 303–344.Google Scholar
[15] Kobayashi, S., Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, 70, Springer-Verlag, New York-Heidelberg, 1972.Google Scholar
[16] Tamura, M., Gauss maps of surfaces in contact space forms. Comment. Math. Univ. St. Pauli 52(2003), no. 2, 117–123.Google Scholar
[17] Tanaka, N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math. (N.S.) 2(1976), no. 1, 131–190.Google Scholar
[18] Tanno, S., Sur une variété de K-contact métrique de dimension 3 . C. R. Acad. Sci. Paris Sér. A-B 263(1966), A317–A319.Google Scholar
[19] Tanno, S., Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314(1989), no. 1, 349–379. doi:10.1090/S0002-9947-1989-1000553-9Google Scholar
[20] Thurston, W. P., Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997.Google Scholar
[21] Vranceanu, G., Leçons de géométrie différentielle. Vol. 1, Ed. Acad. Rep. Pop. Roum., Bucarest, 1947.Google Scholar
[22] Webster, S. M., Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13(1978), no. 1, 25–41.Google Scholar