m-quasi Einstein Metric and Paracontact Geometry

Yıl 2022, Cilt: 15 Sayı: 2, 304 - 312, 31.10.2022

Krishnendu De U.c. De Fatemah Mofarreh

https://doi.org/10.36890/iejg.1100147

Öz

The object of the upcoming article is to characterize paracontact metric manifolds conceding $m$-quasi Einstein metric. First we establish that if the metric $g$ in a $K$-paracontact manifold is the $m$-quasi Einstein metric, then the manifold is of constant scalar curvature. Furthermore, we classify $(k,\mu)$-paracontact metric manifolds whose metric is $m$-quasi Einstein metric. Finally, we construct a non-trivial example of such a manifold.

Anahtar Kelimeler

$K$-paracontact metric manifolds, $(k,\mu)$-paracontact metric manifolds, $m$-quasi Einstein metric

Kaynakça

• [1] Blaga, A.M.: η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20, 1-13 (2015).
• [2] Calvaruso, G.: Homogeneous paracontact metric three-manifolds, Illinois J. Math. 55, 697-718 (2011).
• [3] Calviño-Louza, E., Seone-Bascoy, J., Vázquez-Abal, M.E. and Vázquez-Lorenzo, R.: Three-dimensional homogeneous Lorentzian Yamabe solitons, Abh. Math. Semin. Univ. Hambg. 82, 193-203 (2012).
• [4] Cappelletti-Montano, B., Erken, I.K. and Murathan, C.: Nullity conditions in paracontact geometry, Differential Geom. Appl., 30, 665-693 (2012).
• [5] Case, J.: On the non-existence of quasi-Einstein metrics, Pacific J. Math. 248, 227-284 (2010).
• [6] Case, J., Shu, Y. and Wei, G.: Rigidity of quasi-Einstein metrics, Diff. Geom. Appl. 29, 93-100 (2011).
• [7] Chen, X.: Quasi-Einstein structures and almost cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 114, 72 (2020).
• [8] Catino, G.: Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271, 751-756 (2012).
• [9] De, K. and De, U.C.: A note on gradient solitons on para-Kenmotsu manifolds, Int. J.Geom. Methods Mod. Phys. 18, 2150007 (2021).
• [10] De, K. and De, U.C.: Almost quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons in paracontact geometry, Quaestiones Mathematicae.44, 1429-1440 (2021).
• [11] De, K. and De, U.C.: δ-almost Yamabe solitons in paracontact metric manifolds, Mediterr. J. Math. 18, 218 (2021).
• [12] Ghosh, A.: Quasi-Einstein contact metric manifolds, Glasgow Math. J. 57, 569-577 (2015).
• [13] Huang, G. and Wei, Y.: The classification of (m, ρ)-quasi-Einstein manifolds, Ann. Global Anal. Geom. 44, 269-282 (2013).
• [14] Kaneyuki, S. and Williams, F.L.: Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99, 173-177 (1985).
• [15] Pan, Q. and Liu, X.: A Classification of 3-dimensional Paracontact Metric Manifolds with ϕl = lϕ, J. Math. Res. Appl. 38, 509-522 (2018).
• [16] Sato, I.: On a structure similar to the almost contact structure, Tensor (N.S.) 30, 219-224 (1976).
• [17] Wei, G. and Wylie, W.: Comparison geometry for the Bakry-Emery Ricci tensor, J. Differ. Geom. 83, 337-405 (2009).
• [18] Zamkovoy, S.: Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36, 37-60 (2009).