Previous |  Up |  Next

Article

Keywords:
pseudo-hermitian mean curvature vector fields; proper mean curvature; biharmonic submanifolds; biminimal immersions
Summary:
We classify Hopf cylinders with proper mean curvature vector field in Sasakian 3-manifolds with respect to the Tanaka-Webster connection.
References:
[1] Barros, M., Garay, O. J.: On submanifolds with harmonic mean curvature. Proc. Amer. Math. Soc. 123 (1995), 2545–2549. DOI 10.1090/S0002-9939-1995-1254831-7 | MR 1254831 | Zbl 0827.53015
[2] Chen, B. Y.: Some classification theorems for submanifolds in Minkowski space–time. Arch. Math. (Basel) 62 (1994), 177–182. DOI 10.1007/BF01198672 | MR 1255641 | Zbl 0816.53035
[3] Chen, B. Y.: Submanifolds in de Sitter space–time satisfying $\Delta H = \lambda H$. Israel J. Math. 91 (1995), 373–391. MR 1348323 | Zbl 0873.53041
[4] Chen, B. Y.: Report on submanifolds of finite type. Soochow J. Math. 22 (1996), 117–337. MR 1391469 | Zbl 0867.53001
[5] Cho, J. T., Inoguchi, J., Lee, J.-E.: Affine biharmonic submanifolds in 3–dimensional pseudo–Hermitian geometry. Abh. Math. Sem. Univ. Hamburg 79 (2009), 113–133. DOI 10.1007/s12188-008-0014-8 | MR 2541346 | Zbl 1180.58010
[6] Defever, F.: Hypersurfaces of $E^4$ satisfying $\Delta H = \lambda H$. Michigan Math. J. 44 (1997), 355–364. MR 1460420
[7] Defever, F.: Hypersurfaces of $E^4$ with harmonic mean curvature vector. Math. Nachr. 196 (1998), 61–69. DOI 10.1002/mana.19981960104 | MR 1657990
[8] Defever, F.: Theory of semisymmetric conformally flat and biharmonic submanifolds. Balkan J. Geom. Appl. 4 (1999), 19–30. MR 1751643 | Zbl 0980.53006
[9] Dimitric, I.: Submanifolds of $E^m$ with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20 (1992), 53–65. MR 1166218
[10] Ferrández, A., Lucas, P., Meroño, M. A.: Biharmonic Hopf cylinders. Rocky Mountain J. Math. 28 (1998), 957–975. DOI 10.1216/rmjm/1181071748 | MR 1656996
[11] Garay, O. J.: A classification of certain 3–dimensional conformally flat Euclidean hypersurfaces. Pacific J. Math. 162 (1994), 13–25. DOI 10.2140/pjm.1994.162.13 | MR 1247141 | Zbl 0791.53026
[12] Hasanis, Th., Vlachos, Th.: Hypersurfaces in $E^4$ with harmonic mean curvature vector field. Math. Nachr. 172 (1995), 145–169. DOI 10.1002/mana.19951720112 | MR 1330627
[13] Inoguchi, J.: Submanifolds with harmonic mean curvature vector field in contact 3-manifolds. Colloq. Math. 100 (2004), 163–179. DOI 10.4064/cm100-2-2 | MR 2107514 | Zbl 1076.53065
[14] Inoguchi, J.: Biminimal submanifolds in 3–dimensional contact manifolds. Balkan J. Geom. Appl. 12 (1) (2007), 56–67. MR 2321968
[15] Inoguchi, J., Lee, J.-E.: Almost contact curves in normal almost contact $3$-manifolds. submitted.
[16] Inoguchi, J., Lee, J.-E.: Biminimal curves in $2$–dimensional space forms. submitted.
[17] Lee, J.-E.: On Legendre curves in contact pseudo–Hermitian 3–manifolds. Bull. Austral. Math. Soc. 81 (1) (2010), 156–164. DOI 10.1017/S0004972709000872 | MR 2584930 | Zbl 1185.53048
[18] Loubeau, E., Montaldo, S.: Biminimal immersions. Proc. Edinburgh Math. Soc. (2) 51 (2008), 421–437. MR 2465916 | Zbl 1144.58010
[19] Ogiue, K.: On fiberings of almost contact manifolds. Kōdai Math. Sem. Rep. 17 (1965), 53–62. DOI 10.2996/kmj/1138845019 | MR 0178428 | Zbl 0136.18101
[20] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. DOI 10.1307/mmj/1028999604 | MR 0200865
[21] Tanaka, N.: On non–degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math. (N.S.) 2 (1) (1976), 131–190. MR 0589931 | Zbl 0346.32010
[22] Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314 (1989), 349–379. DOI 10.1090/S0002-9947-1989-1000553-9 | MR 1000553 | Zbl 0677.53043
[23] Webster, S. M.: Pseudohermitian structures on a real hypersurface. J. Differential Geom. 13 (1978), 25–41. MR 0520599
Partner of
EuDML logo