Abstract
We study almost contact curves in normal almost contact metric 3-manifolds satisfying \({\triangle{H} = \lambda{H}}\) or \({\triangle^\bot {H} = \lambda{H}}\) . Moreover we study almost contact curve of type AW(k) in normal almost contact metric 3-manifolds. We give natural equations of planar biminimal curves.
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Arroyo J., Barros M., Garay O.J.: A characterisation of helices and Cornu spirals in real space forms. Bull. Austral. Math. Soc. 56, 37–49 (1997)
Arslan, K., Özgür, C.: Curves and surfaces of AW(k) type. In: Defever, F. et al. (eds.) Geometry and Topology of Submanifolds, IX (Valenciennes/Lyon/Leuven, 1997), pp. 21–26. World Scientific Publishing, River Edge (1999)
Arslan K., West A.: Product submanifolds with pointwise 3-planar normal sections. Glasgow Math. J. 37, 73–81 (1995)
Baikoussis C., Blair D.E.: On Legendre curves in contact 3-manifolds. Geom. Dedicata 49, 135–142 (1994)
Barros M., Garay O.J.: On submanifolds with harmonic mean curvature. Proc. Am. Math. Soc. 123, 2545–2549 (1995)
Blair D.E.: On the non-existence of flat contact metric structure. Tohoku Math. J. 28, 373–379 (1976)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Progress in Mathematics, vol. 203. Birkhäuser, Basel (2010)
Chen B.Y.: Submanifolds with planar normal sections. Soochow J. Math. 7, 19–24 (1981)
Chen B.Y.: Differential geometry of submanifolds with planar normal sections. Ann. Mat. Pura Appl. 130(4), 59–66 (1982)
Chen B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)
Chen B.Y.: Some classification theorems for submanifolds in Minkowski space-time. Arch. Math. (Basel) 62, 177–182 (1994)
Ferrández A., Lucas P., Meroño M.A.: Biharmonic Hopf cylinders. Rocky Mountain J. Math. 28, 957–975 (1998)
Inoguchi J.: Submanifolds with harmonic mean curvature vector field in contact 3-manifolds. Colloq. Math. 100, 163–179 (2004)
Inoguchi J., Lee J.-E.: Submanifolds with harmonic mean curvature in pseudo-Hermitian geometry. Archiv. Math. (Brno) 48, 15–26 (2012)
Janssens D., Vanhecke L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)
Kenmotsu K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)
Lee J.-E.: On Legendre curves in contact pseudo-Hermitian 3-manifolds. Bull. Austral. Math. Soc. 81, 156–164 (2010)
Loubeau E., Montaldo S.: Biminimal immersions. Proc. Edinb. Math. Soc. 51(2), 421–437 (2008)
Rukimbira P.: A characterization of flat contact metric geometry. Houston J. Math. 24, 409–414 (1998)
Olszak Z.: Normal almost contact manifolds of dimension three. Ann. Pol. Math. 47, 42–50 (1986)
Özgür C., Tripathi M.M.: On Legendre curves in α-Sasakian manifolds. Bull. Malays. Math. Sci. Soc. 31(2), 91–96 (2008)
Spivak, M.,: A Comprehensive Introduction to Differential Geometry, vol. IV, 2nd edn. Publish or Perish, Wilmington (1979)
Thurston, W.M.: Three-dimensional Geometry and Topology I. Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)
Tripathi M.M.: A note on certain Legendre curves in a Kenmotsu manifold. Ganita 51, 57–58 (2000)
Wełyczko J.: On Legendre curves in 3-dimensional normal almost contact metric manifolds. Soochow J. Math. 33, 929–937 (2007)
Wełyczko J.: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Result. Math. 54, 377–387 (2009)
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Inoguchi, Ji., Lee, JE. Almost contact curves in normal almost contact 3-manifolds. J. Geom. 103, 457–474 (2012). https://doi.org/10.1007/s00022-012-0134-2
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DOI: https://doi.org/10.1007/s00022-012-0134-2