Nagaraja H. G., Kiran Kumar D. L., Prakasha D. G.

$D_{a}$-Homothetic Deformation and Ricci Solitons in $(k, \mu)-$ Contact Metric Manifolds

In this paper, we study $(k,\mu)$-contact metric manifold under $D_a$-homothetic deformation. It is proved that a $D_3$-homothetic deformed locally symmetric $(1, -4)$-contact metric manifold is a Sasakian manifold and the Ricci soliton is shrinking. Further, $\xi^*$-projectively flat and $h$-projectively semisymmetric $(k, \mu)$-contact metric manifolds under $D_a$-homothetic deformation are studied and obtained interesting results.

Keywords:

$D_{a}$-homothetic deformation, Ricci solitons, projective curvature tensor $D_{a}$-homothetic deformation, Ricci solitons, projective curvature tensor,

PDF

___

[1] D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509. Springer Verlag, New York, 1973.
[2] D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, BirkhauserBoston. Inc., Boston, 2002.
[3] D.E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 29 (1977), 319-324.
[4] E. Boeckx, A full classification of contact metric (k;m)- spaces, Illinois J. Math, 44 (2000), 212-219.
[5] J.T. Cho, A conformally flat (k;m)-space, Indian J. Pure Appl. Math. 32 (2001), 501-508.
[6] U.C. De, Y.H. Kim and A.A. Shaikh, Contact metric manifolds with x belonging to (k;m)-nullity distribution, Indian J. Math., 47 (2005), 1-10.
[7] U.C. De, and A. Sarkara, On the quasi-conformal curvature tensor of a (k;m)-contact metric manifold, Math. Reports 14(64), 2 (2012), 115-129.
[8] A. Ghosh, T. Koufogiorgos and R. Sharma, Conformally flat contact metric manifolds, J. Geom., 70 (2001), 66-76.
[9] A. Ghosh and R. Sharma, A classification of Ricci solitons as (k;m)-contact metrics, Springer Proceedings in Mathematics and Statistics, Springer Japan, (2014), 349-358.
[10] R.S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics, 71 (1988), 237-262.
[11] T. Ivey, Ricci solitons on compact 3-manifolds, Differential Geom. Appl. 3 (1993), 301-307.
[12] J.B. Jun, A. Yildiz and U.C. De, On f-recurrent (k;m)-contact metric manifolds. Bulletin of the Korean Mathematical Society, 45(4) (2008), 689-700.
[13] P. Majhi and G. Ghosh, Concircular vectors field in (k;m)-contact metric manifolds. International Electronic Journal of Geometry, 11(1) (2018), 52-56.
[14] B.J. Papantoniou, Contact Riemannian manifolds satisfying R(X;x ) R = 0 and x 2 (k;m)-nullity distribution, Yokohama Math. J., 40 (1993), 149-161.
[15] D.G. Prakasha, C.S. Bagewadi and Venkatesha, On pseudo projective curvature tensor of a contact metric manifold, SUT J. Math. 43 (2007), 115-126.
[16] R. Sharma, Certain results on K-contact and (k;m)-contact metric manifolds, J. Geom., 89 (2008), 138-147.
[17] R. Sharma and T. Koufogiorgos, Locally symmetric and Ricci symmetric contact metric manifolds, Ann. Global Anal. Geom., 9 (1991), 177-182.
[18] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Mathematical Journal, Second Series, (40(3) (1988), 441-448.
[19] S. Tanno, The topology of contact Riemannian manifolds, Illinois Journal of Mathematics, 12(4) (1968), 700-717.
[20] M.M. Tripathi, Ricci solitons in contact metric manifolds, arXiv:0801.4222v1 [math.DG], 2008.
[21] M.M. Tripathi and. J.S. Kim, On the concircular curvature tensor of a (k;m)-manifold, Balkan J. Geom. Appl. 9(1) (2004), 114-124.
[22] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, World Scientific publishing, Singapore, 3 (1984).