Some conformal vector fields and conformal Ricci solitons on N(k)-contact metric manifolds

Document Type : Original Article


1 Department of Pure Mathematics, Faculty of Science, University of Calcutta, Kolkata, India

2 Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal, India

3 Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal



The target of this paper is to study N(k)-contact metric manifolds with some types of conformal vector fields like φ-holomorphic planar conformal vector fields and Ricci biconformal vector fields. We also characterize N(k)-contact metric manifolds allowing conformal Ricci almost soliton. Obtained results are supported by examples.


Main Subjects

[1] G. P. Alfonso, M. M. S. Jose, Bi-conformal vector fields and their applications, arXiv:math-ph/0311014v2.
[2] C. Baikoussis, D. E. Blair, T. Koufogiorgos, A decomposition of the curvature tensor of a contact manifold satisfying R(X, Y )ξ = k(η(Y )X − η(X)Y ), Math. Technical Reports, University of Ioannina, Greece, 1992.
[3] N. Basu, A. Bhattacharyya, Conformal Ricci soliton in Kenmotsu manifold, Global J. Adv. Res. on Class. Mod. Geom., 4 (2015) 15-21.
[4] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math., 203, Birkhauser, 2010.
[5] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91 (1995) 189-214.
[6] H. D. Cao, B. Chow, Recent developments on the Ricci flow, Bull. Amer. Math. Soc., 36 (1999) 59-74.
[7] U. C. De, Certain results on N(k)-contact metric manifolds, Tamkang J. Math., 49 (2018) 205-220.
[8] U. C. De, A. Yildiz, S. Ghosh, On a class of N(k)-contact metric manifolds, Math. Reports, 16 (2004) 207-217.
[9] S. Deshmukh, Geometry of conformal vector fields, Arab J. Math. Sci, 23 (2017) 44-73.
[10] A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav., 21 (2004) 171-218.
[11] A. Ghosh, Holomorphically planar conformal vector fields on contact metric manifolds, Acta Math. Hungarica, 129 (2010) 357-367.
[12] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986, 237- 262.) Contemp. Math. 71, American Math. Soc., 1988.
[13] J. B. Jun, U. C. De, On N(k)-contact metric manifolds satisfying some curvature conditions, Kyungponk Math. J., 34 (2011) 457-468.
[14] H. G. Nagaraja, K. Venu, f-Kenmotsu metric as conformal Ricci soliton, Ann. Univ. Vest. Timisoara. Ser. Math. Inform., LV (2017) 119-127.
[15] M. Okumura, Some remarks on space with certain contact structures, Tohoku Math. J., 14 (1962) 135-145.
[16] M. Okumura, On infinitesimal conformal and projective transformations of normal contact spaces, Tohoku Math. J., 14 (1962),398-412.
[17] C. Ozgur, S. Sular, On N(k)-contact metric manifolds satisfying certain conditions, SUT J. Math., 44 (2008) 89-99.
[18] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1.
[19] S. Pigola, M. Rigoli, M., Rimoldi, A. G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super Pisa Cl. Sci., 10 (2011) 757-799. [20] R. Sharma, D. E. Blair, Conformal motion of contact manifolds with characteristic vector field in the k-nullity distribution, Illinois J. Math., 40 (1996) 553-563.
[21] R. Sharma, Holomorphically planar conformal vector fields on almost Hermitian manifolds, Contemp. Math., 337 (2003) 145-154.
[22] R. Sharma, L. Vrancken, Conformal classification of (k, µ)-Contact manifolds, Kodai Math. J., 33 (2010) 267-282.
[23] R. Sharma, Certain results on K-contact and (k, µ)-contact metric manifolds, J. Geom., 89 (2008) 138-147.
[24] R. Sharma, Conformal and projective characterizations of an odd dimensional unit sphere , Kodai Math. J., 42 (2019) 160-169.
[25] S. Tanno, Some transformations on manifolds with almost contact and contact metric structures, Tohoku Math. J., 15 (1963) 140-147.
[26] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970