Some Curves on alpha-para Kenmotsu manifolds with semisymmetric metric connections.

Abstract

The object of the present paper is to study biharmonic almost contact curves on three-dimensional alpha-para Kenmotsu manifolds with respect to semisymmetric metric connections. With respect to semisymmetric metric connection slant curves have been analysed. Locally phi-symmetric Legendre curves with respect to semisymmetric metric connections have also been considered. An example is given

Keywords

• Biharmonic curves,Almost contact curves,Slant curves,alpha-para Kenmotsu manifolds,Locally phi-symmetric,Semisymmetric metric connections

References

1. [1] Baikoussis, C. and Blair, D. E., On Legendre curves in contact 3-manifolds Geom. Dedicata, 49(1994), 135-142.
2. [2] Barman, A., On Lorentzian a-Sasakian manifolds admitting a type of semisymmetric metric connection, Novi Sad J. Math. 44(2014), 77-88.
3. [3] Blair, D. E., Contact manifolds in Riemannian Geometry, Lecture notes in Math 509, Springer-Verlag, Berlin-Heidelberg-New York(1976).
4. [4] Blair, D. E., Kim, J. S. and Tripathi, M. M., On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42(2005), 883-892.
5. [5] Caddeo, R., Montaldo, S. and Piu, P., Biharmonic curves on a surface, Rend, Mat. Appl. 21(2001), no.1-4, 143-157.
6. [6] Calin, C. and Crasmareanu, M., Slant curves in 3-dimensional Normal Almost Contact Geometry, Mediterr. J. Math. 10(2013), 1067-1077.
7. [7] Cappelletti-Montano, B., Bi-Legendrian structures and paracontact geometry, Int. J. Geom. Meth. Mod. Phys. 6(2009), 487-504.
8. [8] Chinea, D. and Gonzales, C., A classification of almost contact metric manifolds, Ann. Mat. Pure Appl., 156(1990), 15-36.

9. [9] Cho, J. T., Inoguchi, J. I. and Lee, J. E., On slant curves in Sasakian space forms, J. Korean Math. Soc. 74(2006), 359-367.
10. [10] Cho, J. T. and Lee, J. E., Slant curves in contact Pseudo-Hermitian manifolds, Bull. Austral. Math. Soc., 78(2008), 383-396.
11. [11] Erdem, S., On almost (para) contact (hyperbolic) metric manifolds and harmonicity of (f;f0 )-holomorphic maps between them, Houston J. Math. 28(2002), 21-45.
12. [12] Fetcu, D., Biharmonic Legendre curves in Sasakian space forms, J. Korean Math. Soc. 45(2008), 393-404.
13. [13] Friedmann, A. and Schouten, ¨U berdie Geometrie der halbsymmetrischen ¨U bertragung, Math.Z., 21(1924), 211-223.
14. [14] Hayden, H. A., Subspaces of space with torsion, Proc. London Math. Soc., 34(1932), 27-50.
15. [15] Inoguchi, J. I. and Lee, J. E., Affine biharmonic curves in 3-dimensional homogeneous geometries, Mediterr. J. Math. 10(2013), 571-592.
16. [16] Inoguchi, J. I. and Lee, J. E., On slant curves in normal almost contact metric 3-manifolds, Beitr. Algebra Geom. 55(2014), 603-620.
17. [17] Janssens, D. and Vanhecke, L., Almost contact structures and curvature tensors, Kodai. Math. J. 4(1981), 1-27.
18. [18] Kaneyuki, S. and Williams, F. L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99(1985), 173-187.
19. [19] Majhi, P., A note on a-para Kenmotsu manifolds, Facta Universitatis, 31(2016), 227-236.
20. [20] Montaldo, S. and Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Revista De La Union Matematica Argentina, 47(2006), 1-22.
21. [21] Olszak, Z., Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. 47(1986), 41-50.
22. [22] Oubina, J. A., New classes of alomost contact metric structures, Publ. Math. Debrecen, 32(1985), 187-193.
23. [23] P, Dacko., On almost para-cosymplectic manifolds, Tsukuba J. Math. 28(2004), 193-213.
24. [24] Sarkar, A. and Biswas, D., Legendre curves on three dimensional Heisenberg Groups, Facta Universitatis, 28(2013), 241-248.
25. [25] Sarkar, A., Mondal, A. and Biswas, D., Some curves on three-dimensional trans-Sasakian manifolds with semisymmetric metric connection, Palest. J. Math., 5(2016), 195-203.
26. [26] Sarkar, A. and Mondal, A., Certain curves on some classes of three-dimensional almost contact metric manifolds, Revista de La Union Mat. Argentina, 58(2017), 107-125.
27. [27] Sarkar, A. and Sil, Amit, Legendre curves on three-dimensional quasi-Sasakian manifolds with semisymmetric metric connection, Acta Mathematica Academiae Paedagogiace Nyiregyhaziensis,(to appear, 2017).
28. [28] Sarkar, A. and Sil, Amit, Curves on some classes of Kenmotsu manifolds, Ann. Univ. Sci Budapest, 59(2016), 101-112.
29. [29] Sarkar, A. and Sil, Amit, Certain curves on a-para Kenmotsu manifolds, (communicated)
30. [30] Schouten, J. A.: Ricci Calculus, An Introduction to Tensor Analysis and Its Geometrical Applications, Springer-Verlag, Berlin, 1954
31. [31] Sharfuddin, A. and Hussain, S. I., Semi Symmetric metric connections in almost contact manifolds, Tensor, New Ser. 30(1976), 133-139.
32. [32] Srivastava, K. and Srivastava, S. K., On a class of a-para Kenmotsu manifolds, Mediterr. J. Math. 13(2016), 391-399.
33. [33] Welyczko, J., On Legendre curves in 3-dimensional normal almost contact metric manifolds, Soochow J. Math., 33(2007), 929-937.
34. [34] Welyczko, J., Slant curves in 3-dimensional normal almost paracontact metric manifolds, Mediterr. J. Math, 11(2014), 965-978.
35. [35] Yano, K., On Semi Symmetric connection, Rev. Roum, Math. Pure Appl., 15(1970), 1570-1586.
36. [36] Zamkovoy, S., Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom.36(1)(2009), 37-60.