Georgian Math. J. 2017; aop Research Article Doddabhadrappla G. Prakasha, Luis M. Fernández* and Kakasab Mirji The M-projective curvature tensor őeld on generalized ( κ , μ )-paracontact metric manifolds https://doi.org/10.1515/gmj-2017-0054 Received August 1, 2016; revised January 27, 2017; accepted March 17, 2017 Abstract: We consider generalized (κ , μ)-paracontact metric manifolds satisfying certain ŕatness condi- tions on the M-projective curvature tensor. Speciőcally, we study ξ -M-projectively ŕat and M-projectively ŕat generalized (κ , μ)-paracontact metric manifolds and, further, ϕ-M-projectively symmetric generalized (κ ̸ =−1, μ)-paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail. Keywords: Generalized (κ , μ)-paracontact metric manifold, M-projective curvature tensor, ŕat manifold, η-Einstein manifold MSC 2010: 53C15, 53C25 1 Introduction The study of paracontact geometry was initiated by Sato [21] and Kaneyuki and Williams [13]. Recently, there seems to be an increasing interest in paracontact geometry and, in particular, in the study of para-Sasakian manifolds, due to its links to more consolidated theory of para-Kaehler manifolds and to their role in pseudo- Riemannian geometry and mathematical physics (see, for instance, [8, 9, 11]). Actually, the notion of almost paracontact structure is an analogue of that one of almost contact structure and is closely related to the almost product structure. Moreover, the geometry of paracontact metric manifolds can be related to the theory of Legendre foliations [4, 5]. A systematic study of paracontact metric manifolds and their subclasses was started by Zamkovoy [22]. Subsequently, many geometers have studied paracontact metric manifolds and obtained various important properties of these manifolds (see, for instance, [1ś3, 12, 20, 23]). Paracontact metric manifolds have been studied from diferent points of view. Recently, Cappelletti- Montano and Di Terlizzi [5] have introduced the notion of (κ , μ)-paracontact metric manifolds as those para- contact metric manifolds such that the underlying paracontact metric structure (ϕ, ξ , η, g) satisőes the con- dition R(X, Y)ξ = κ[η(Y)X − η(X)Y ]⋇ μ[η(Y)hX − η(X)hY ] for some real numbers κ and μ, where 2h denotes the Lie derivative of ϕ in the direction of ξ , giving several examples. Later, two of the authors, among others, have studied them in [19]. The class of (κ , μ)-para- contact metric manifold contains para-Sasakian manifolds. Assuming that κ and μ are smooth functions, Doddabhadrappla G. Prakasha, Department of Mathematics, Faculty of Science & Technology, Karnatak University, Pavate Nagar, Dharwad 580 003, India, e-mail: prakashadg@gmail.com *Corresponding author: Luis M. Fernández, Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarőa s/n, 41012 Sevilla, Spain, e-mail: lmfer@us.es Kakasab Mirji, Department of Mathematics, KLS’s Gogte Institute of Technology, Belagavi, India, e-mail: mirjikk@gmail.com Brought to you by | University of Gothenburg Authenticated Download Date | 11/25/17 9:11 AM