SLANT CURVES IN 3-DIMENSIONAL ALMOST CONTACT METRIC GEOMETRY

Year 2015, Volume: 8 Issue: 2, 106 - 146, 30.10.2015

Jun İchi Inoguchı , Ji-eun Lee

https://doi.org/10.36890/iejg.592300

Cited By: 12

Abstract

Keywords

slant curves, canonical connection, normal almost contact manifold

References

• [1] Arslan, K. and Özgür, C., Curves and surfaces of AW(k) type, in: Geometry and Topology of Submanifolds, IX (Valenciennes/Lyon/Leuven, 1997), (F. Defever, F. et al. eds.), World Scientific Publishing Co., Inc., River Edge, 1999, pp. 21-26.
• [2] Arslan, K. and West, A., Product submanifolds with pointwise 3-planar normal sections, Glasgow Math. J. 37(1995), no. 1, 73-81.
• [3] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125(1997), 1503-1509.
• [4] Baikoussis, C. and Blair, D. E., On Legendre curves in contact 3-manifolds, Geom. Dedi- cata 49(1994), 135-142.
• [5] Belkhelfa, M., Dillen, F. and Inoguchi, J., 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.
• [6] Bianchi, L., Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Tereza, Tomo XI(1898), 267-352. English Translation: On the three-dimensional spacse which admit a continuous group of motions, General Relativity and Gravitation 33(2001), no. 12, 2171-2252.
• [7] Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhäuser, Boston, Basel, Berlin, 2002.
• [8] Blair, D. E., Dillen, F., Verstraelen, L. and Vrancken, L., Deformations of Legendre curves, Note Mat. 15(1995), no. 1, 99-110.
• [9] Cabrerizo, J. L., Fern´andez, M. and G´omez, J. S., On the existence of almost contact structure and the contact magnetic field, Acta Math. Hungar. 125(2009) no. 1-2, 191-199.
• [10] Cabrerizo, J. L., Ferna´ndez, M. and G´omez, J. S., The contact magnetic flow in 3D Sasakian manifolds, J. Phys. A: Math. Theor. 42(2009), 19, 195201:1-10.
• [11] C˘alin, C. and Crasmareanu, M., Slant curves in 3-dimensional normal almost contact geometry, Mediterr. J. Math. 10(2013), no. 2, 1067-1077.
• [12] C˘alin, C., Crasmareanu, M. and Munteanu, M.-I., Slant curves in 3-dimensional f - Kenmotsu manifolds, J. Math. Anal. Appl. 394(2012), no. 1, 400-407.
• [13] C˘alin, C. and Ispas, M., On a normal contact metric manifold, Kyoungpook Math. J. 45(2005), 55-65.
• [14] Camcı, C., Extended cross product in a 3-dimensional almost contact metric manifold with application to curve theory, Turkish J. Math. 35(2012), 1-14.
• [15] Cartan, E., Le¸con sur la geometrie des espaces de Riemann, Second Edition, Gauthier- Villards, Paris, 1946.
• [16] Cho, J. T., On some classes of almost contact metric manifolds, Tsukuba J. Math. 19(1995), no. 1, 201-217.
• [17] Cho, J. T., Inoguchi, J. and Lee, J.-E., On slant curves in Sasakian 3-manifolds, Bull. Austral. Math. Soc. 74(2006), 359-367.
• [18] Cho, J. T., Inoguchi, J. and Lee, J.-E., Biharmonic curves in 3-dimensional Sasakian space forms, Annali di Mat. Pura Appl. 186(2007), no. 4, 685-701.
• [19] 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), 113-133.
• [20] Cho, J. T., and Lee, J.-E., Slant curves in contact pseudo-Hermitian 3-manifolds, Bull. Aust. Math. Soc. 78(2008), no. 3, 383-396.
• [21] Dacko, P., On almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distributions, Balkan J. Geom. Appl. 5(2000), 47-60.
• [22] Dileo, G., A classification of certain almost α-Kenmotsu manifolds, Kodai Math. J. 34(2011), 426–445.
• [23] Dileo, G. and Pastore, A. M., Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14(2007), 343-354.
• [24] Dileo, G. and Pastore, A. M., Almost Kenmotsu manifolds with a condition of η- parallelism, Differential Geom. Appl. 27(2009), 671-679.
• [25] Dileo, G. and Pastore, A. M., Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93(2009), 46-61.
• [26] Dru¸ta˘-Romaniuc, S. L., Inoguchi, J., Munteanu, M.-I. and Nistor, A. I., Magnetic curves in Sasakian manifolds, J. Nonlinear Math. Phys., to appear.
• [27] Ferrandez, A., Riemannian versus Lorentzian submanifolds, some open problems, in: Proc. Workshop on Recent Topics in Differential Geometry, Santiago de Compostera, Depto. Geom. y Topolog´ıa, Univ. Santiago de Compostera 89(1998), 109-130.
• [28] Ferrandez, A., Gimenez, A. and Lucas, P., Null helices in Lorentzian space forms, Int. J. Mod. Phys. A 16(2001), 4845-4863.
• [29] Ferrandez, A., Gimenez, A. and Lucas, P., Null generalized helices in Lorentz-Minkowski spaces, J. Phys. A. 35(2002), no. 39, 8243-8251.
• [30] Gluck, H., Geodesics in the unit tangent bundle of a round sphere, L’Enseignement Math´ematique 34(1988), 233-246.
• [31] Goldberg, S. I. and Yano, Y., Integrability of almost cosymplectic structure, Pacific J. Math. 31(1969), 373-382.
• [32] Güven¸c, S¸. and Özgür, C., On slant curves in trans-Sasakian manifolds, Rev. Un. Mat. Argentina 55(2014), no. 2, 81-100.
• [33] Hua, Z.-H. and Sun, L., Slant curves in the unit tangent bundles of surfaces, ISRN Ge- ometry 2013(2013), Article ID 821429, 5 pages.
• [34] Inoguchi, J., Minimal surfaces in 3-dimensional solvable Lie groups, Chinese Ann. Math. B. 24(2003), 73-84.
• [35] Inoguchi, J., Submanifolds with harmonic mean curvature vector field in contact 3- manifolds, Coll. Math. 100(2004), 163-179.
• [36] Inoguchi, J., Minimal surfaces in 3-dimensional solvable Lie groups II, Bull. Austral. Math. Soc. 73(2006), 365-374.
• [37] Inoguchi, J. and J.-E. Lee, Submanifolds with harmonic mean curvature in pseudo- Hermitian geometry, Archiv. Math. (Brno) 48(2012), 15-26.
• [38] Inoguchi, J. and Lee, J.-E., Biminimal curves in 2-dimensional space forms, Commun. Korean Math. Soc. 27(2012), no. 4, 771-780.
• [39] Inoguchi, J. and Lee, J.-E., Almost contact curves in normal almost contact metric 3- manifolds, J. Geom. 103(2012), 457-474.
• [40] Inoguchi, J. and Lee, J.-E., Affine biharmonic curves in 3-dimensional homogeneous ge- ometries, Mediterr. J. Math. 10(2013), no. 1, 571-592.
• [41] Inoguchi, J. and Lee, J.-E., On slant curves in normal almost contact metric 3-manifolds, Beitr¨age Algebra Geom. 55(2014), no. 2, 603-620.
• [42] Inoguchi, J. and Lee, J.-E., Slant curves in 3-dimensional almost f -Kenmotsu manifolds, in preparation.
• [43] Inoguchi, J., and Lee, S., A Weierstrass representation for minimal surfaces in Sol., Proc. Amer. Math. Soc. 136(2008), 2209-2216.
• [44] Inoguchi, J., and Lee, S., Null curves in Minkowski 3-space, Internat. Electr. J. Geom. 1(2008), no. 2, 40-83.
• [45] Inoguchi, J., and Munteanu, M. I., Periodic magnetic curves in Berger spheres, Tˆohoku Math. J, to appear.
• [46] Jun, J.-B. Kim, I. B. and Kim,U. K., On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34(1994), no. 2, 293-301.
• [47] Kim, T. W.and Pak, H. K., Canonical foliations of certain classes of almost contact metric structure, Acta Math. Sinica (Eng. Ser.) 21(2005), no. 4, 841-846.
• [48] Kobayashi, S., Transformation Groups in Differential Geometry, Ergebnisse der Mathe- matik und Ihere Grenzgebiete, 70, Springer Verlag, 1972.
• [49] Kobayashi, S. and Nomizu, K., Foundations of Differentail Geometry II, Interscience Tracts in Pure and Applied Math. 15, Interscience Publishers, 1969.
• [50] Koto, S. and Nagao, M., On an invariant tensor under a CL-transformation, Kodai Math. Sem. Rep. 18(1966), 87-95.
• [51] Lancret, M. A., M´emoire sur les courbes `a double courbure, M´emoires pr´esent´es `a l’Institut1(1806), 416-454.
• [52] Lee, J.-E., On Legendre curves in contact pseudo-Hermitian 3-manifolds, Bull. Aust. Math. Soc. 81(2010), no. 1, 156-164.
• [53] Lee, J.-E., Suh, Y. J. and Lee, H., C-parallel mean curvature vector fields along slant curves in Sasakian 3-manifolds, Kyungpook Math. J. 52(2012), 49-59.
• [54] Montaldo, S. and Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47(2006), no. 2, 1-22.
• [55] Okumura, M., Some remarks on space[s] with a certain contact structure, Tˆohoku Math. J. 14(1962), 135-145.
• [56] Olszak, Z., On almost cosymplectic manifolds, K¯odai Math. J. 4(1981), 229-250.
• [57] Olszak, Z., Normal almost contact manifolds of dimension three, Ann. Pol. Math. 47(1986), 42-50.
• [58] Olszak, Z., Locally conformal almost cosymplectic manifolds, Coll. Math. 57(1989), no. 1, 73-87.
• [59] Olszak, Z. and Ro¸sca, R., Normal locally conformal almost cosymplectic manifolds, Publ. Math. Debrecen 39(1991), no. 3-4, 315-323.
• [60] O’Neill, B., Elementary Differential Geometry, Academic Press, 1966.
• [61] Özgür, C. and Guüvenç, S¸., On some types of slant curves in contact pseudo-Hermitian 3-manifolds, Ann. Pol. Math. 104(2012), 217-228.
• [62] Saltarelli, V., Three-dimensional almost Kenmotsu manifolds satisfying certain nullity conditions, Bull. Malays. Math. Sci. Soc. 38(2015), no. 2, 437-459.
• [63] Sasahara, T., A short survey of biminimal Legendrian and Lagrangian submanifolds, Bull. Hachinohe Inst. Tech. 28(2009), 305-315. (http://ci.nii.ac.jp/naid/110007033745/)
• [64] Sasaki, S. and Hatakeyama, Y., On differentiable manifolds with certain structures which are close related to almost contact structure II, Tˆohoku Math. J. 13(1961), 281-294.
• [65] Smoczyk, K., Closed Legendre geodesics in Sasaki manifolds, New York J. Math. 9(2003), 23-47.
• [66] Spivak, M., A Comprehensive Introduction to Differential Geometry IV (3rd edition), Publish or Perish, 1999.
• [67] Srivastava, S.K., Almost contact curves in trans-Sasakian 3-manifolds, preprint, arXiv:1401.6429v1[math.DG].
• [68] Struik, D. J., Lectures on Classical Differential Geometry, Addison-Wesley Press Inc., Cambridge, Mass., 1950, Reprint of the second edition, Dover, New York, 1988.
• [69] Takamatsu, K. and Mizusawa, H., On infinitesimal CL-transformations of compact normal contact metric spaces, Sci. Rep. Niigata Univ. Ser. A. 3(1966), 31-39.
• [70] Tamura, M., Gauss maps of surfaces in contact space forms, Comm. Math. Univ. Sancti Pauli 52(2003), 117-123.
• [71] Tanaka, N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan con- nections, Japan. J. Math. 2(1976), 131-190.
• [72] S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314(1989), 349-379
• [73] Tashiro, Y. and Tachibana, S. I., On Fubinian and C-Fubinian manifolds, K¯odai Math. Sem. Rep. 15(1963), 176-183.
• [74] Thurston, W. M., Three-dimensional Geometry and Topology I, Princeton Math. Series., vol. 35 (S. Levy ed.), 1997.
• [75] Tricerri, F. and Vanhecke, L., Homogeneous Structures on Riemannian Manifolds, Lecture Notes Series, London Math. Soc. 52, (1983), Cambridge Univ. Press.
• [76] Vranceanu, G., Le¸cons de G´eom´etrie Diff´erentielle I, Ed. Acad. Rep. Pop. Roum., Bu- carest, 1947.
• [77] Webster, S. M., Pseudohermitian structures on a real hypersurface, J. Differ. Geom. 13(1978), 25-41.
• [78] We-lyczko, J., On Legendre curves in 3-dimensional normal almost contact metric mani- folds, Soochow J. Math. 33(2007), no. 4, 929-937.
• [79] We-lyczko, J., On Legendre curves in 3-dimensional normal almost paracontact metric manifolds, Result. Math. 54(2009), 377-387.
• [80] Yanamoto, H., C-loxodrome curves in a 3-dimensional unit sphere equipped with Sasakian structure, Ann. Rep. Iwate Med. Univ. School of Liberal Arts and Science 23(1988), 51-64.