ON PSEUDO-SYMMETRY CURVATURE CONDITIONS OF GENERALIZED $(k,\mu)$-PARACONTACT METRIC MANIFOLDS

Year 2017, Volume: 5 Issue: 2, 239 - 247, 15.10.2017

Sourav Makhal U. C. De

Abstract

In this paper we investigate Ricci pseudo-symmetric and  Ricci generalized pseudo-symmetric generalized $(k,\mu )$-paracontact metric manifolds. Besides this we characterize generalized $(k,\mu )$-paracontact metric manifolds satisfying the curvature conditions $Q(S,R)=0$ and $Q(S,g)=0$, where $S$, $R$ are the Ricci tensor and curvature tensor respectively. Several corollaries are also obtained.

Keywords

Generalized $(k, \mu)$-paracontact metric manifold, Ricci pseudo-symmetric manifold, Ricci generalized pseudo-symmetric manifold, Einstein manifold

References

• [1] Blair, D.E., Koufogiorgos, T. and Papatoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91(1995), 189-214.
• [2] Calvaruso. G., Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55(2011), 697-718.
• [3] Calvaruso, G. and A. Zaeim, A complete classication of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces, J. Geom. Phys, 80(2014), 15-25.
• [4] Calvaruso, G. and Martin-Molina. V., Paracontact metric structure on the unit tangent sphere bundle, Ann. Math. Pura Appl.194(2015), 1359-1380.
• [5] Calvaruso, G. and Perrone, A., Ricci solitons in three-dimensional paracontact geometry, J. Geom. Phys, 98(2015), 1-12.
• [6] Capplelletti-Montano, B., Kupeli Erken, I and Murathan, C., Nullity conditions in paracon- tact geometry, Diff. Geom. Appl. 30(2012), 665-693.
• [7] Cappelletti-Montano, B., Carriazo, A., Martin-Molina, V., Sasaki-Einstein and paraSasaki- Einstein metics from $(k,\mu )$-structure, J. Geom. Phys, 73(2013), 20-36.
• [8] Cappelletti-Montano, B. and Di Terlizzi, L., Geometric structure associated to a contact metric $(k,\mu )$-space, Pacic J. Math., 246(2010), 257-292.
• [9] De, U.C., Han, Y. and Mandal, K., On para-sasakian manifolds satisfying certain Curvature Conditions, Filomat 31(2017), 1941-1947.
• [10] De, U.C., Deshmukh, S. and Mandal, K., On three-dimensional N(k)-paracontact metric manifolds and Ricci solitons, to appear in Bull. Iranian Math. Soc.
• [11] De, U.C. and Pathak, G., On 3-dimensional Kenmotsu manifolds, Indian J. Pure Appl. Math.,35(2004), 159-165.
• [12] Jun, J.-B. and Kim, U.-K., On 3-dimensional almost contact metric manifolds, Kyungpook Math. J., 34(1994), 293-301.
• [13] Kowalczyk, D., On some subclass of semi-symmetric manifolds, Soochow J. Math.,27(2001), 445-461.
• [14] Kupeli Erken, I., Generalized $(\tilde k\neq-1,\tilde\mu)$-paracontact metric manifolds with $\xi(\tilde\mu)=0,$; Int. Electron. J. Geom., 8(2015), 77-93.
• [15] Kupeli Erken, I. and Murathan, C., A complete study of three-dimensional paracontact $(k,\mu,\nu)$-spaces, arXiv: 1305.1511.
• [16] Kaneyuki, S. and Williams, F.L, Almost paracontact and parahodge structure on manifolds, Nagoya Math. J. 99(1985), 173-187.
• [17] L. Verstraelen, Comments on pseudo-symmetry in sense of R. Deszcz, in: Geometry and Topology of submanifolds, World Sci. Publication. 6(1994), 199-209.
• [18] Szabo, Z. I., Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$ the local version, J. Diff. Geom. 17(1982), 531-582.
• [19] Zamkovoy, S., Canonical connection on paracontact manifolds, Ann. Global Anal. Geom. 36(2009), 37-60.