Parabolic Geodesics in Sasakian 3-Manifolds

Cho, Jong Taek ; Inoguchi, Jun-ichi ; Lee, Ji-Eun

Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 396-410

Résumé

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

Détail

DOI : 10.4153/CMB-2011-035-2

Mots-clés : 58E20, parabolic geodesics, pseudo-Hermitian geometry, Sasakian manifolds

Bibliographie

[1] [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] [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] [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] [4] Blair, D. E., Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics, 509, Springer-Verlag, Berlin-New York, 1976. Google Scholar

[5] [5] Blair, D. E., Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, 203, Birkhäuser Boston Inc., Boston, MA, 2002. Google Scholar

[6] [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] [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] [8] Cartan, E., Leçons sur la géométrie des espaces de Riemann. Second Ed., Gauthier-Villards, Paris, 1946. Google Scholar

[9] [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/S0004972700040429 Google Scholar

[10] [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-x Google Scholar

[11] [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/s12188-008-0014-8t Google Scholar

[12] [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/S0004972708000737 Google Scholar

[13] [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.3160270403 Google Scholar

[14] [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] [15] Kobayashi, S., Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, 70, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar

[16] [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] [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] [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] [19] Tanno, S., Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314(1989), no. 1, 349–379. doi:10.1090/S0002-9947-1989-1000553-9 Google Scholar

[20] [20] Thurston, W. P., Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997. Google Scholar

[21] [21] Vranceanu, G., Leçons de géométrie différentielle. Vol. 1, Ed. Acad. Rep. Pop. Roum., Bucarest, 1947. Google Scholar

[22] [22] Webster, S. M., Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13(1978), no. 1, 25–41. Google Scholar