Ratio Mathematica Volume 48, 2023 W8 - Curvature tensor in generalized Sasakian-space-forms Gyanvendra Pratap Singh* Rajan† Anand Kumar Mishra‡ Pawan Prajapati§ Abstract The generalized Sasakian-space-forms and their properties have been examined by various researchers such as Alegre and Carriazo [2008], Prakasha [2012], Sarkar and Akbar [2014], Shanmukha et al. [2018], Sarkar and Sen [2012], Rajan and Singh [2020] and Sarkar and Sen [2012]. Motivated by the results of these works, we have proposed the idea of the W8−curvature tensor in generalized Sasakian-space- forms. The main goal of this paper is to investigate the curvature properties of generalized Sasakian-space-forms that satisfy the con- ditions ξ − W8− flatness, ϕ − W8−semi-symmetric, W8 · Q = 0, W8 · R = 0 and to prove some interesting results. Keywords: Sasakian-space-form, generalized Sasakian-space-form, ϕ−recurrent, ϕ−symmetric, ϕ−semi-symmetric, W8− curvature ten- sor, Einstein manifold, η−Einstein manifold. 2020 AMS subject classifications:53C15, 53C25, 53D15 1 *Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009 (UP) INDIA; gpsingh.singh700@gmail.com. †Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009 (UP) INDIA; rajanvishwakarma497@gmail.com. ‡Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009 (UP) INDIA; aanandmishra1796@gmail.com. §Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009 (UP) INDIA; pawanpra123@gmail.com. 1Received on November 19, 2022 . Accepted on July 5, 2023. Published on August 1, 2023. DOI: 10.23755/rm.v39i0.961. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Gyanvendra Pratap Singh, Rajan, Anand Kumar Mishra, Pawan Prajapati 1 Introduction The nature of a Riemannian manifold depends on the curvature tensor R of the manifold. It is well known that the sectional curvatures of a manifold determine its curvature tensor completely. Real space-forms are Riemannian manifolds with constant sectional curvature c, and their curvature tensor is given by R(X, Y )Z = c{g(Y, Z)X − g(X, Z)Y }. Representation for these spaces are hyperbolic spaces (c < 0), spheres (c > 0) and Euclidean spaces (c = 0). The Sasakian manifold determines the ϕ-sectional curvature of a Sasakian-space- form, and it has a specific form for its curvature tensor. The Kenmotsu and cosym- plectic space-forms use the same notation. Alegre et al. [2004] developed and researched generalized Sasakian-space-forms in an effort to generalize such space -forms in a shared frame. A generalized Sasakian-space-form is an almost contact metric manifold (M2n+1, ϕ, ξ, η, g), whose curvature tensor is given by R(X, Y )Z = f1{g(Y, Z)X − g(X, Z)Y } + f2{g(X, ϕZ)ϕY − g(Y, ϕZ)ϕX + 2g(X, ϕY )ϕZ} + f3{η(X)η(Z)Y − η(Y )η(Z)X + g(X, Z)η(Y )ξ − g(Y, Z)η(X)ξ}. The Riemannian curvature tensor of a generalized Sasakian-space-form M2n+1(f1, f2, f3) is simply given R = f1R1 + f2R2 + f3R3, where f1, f2, f3 are differential functions on M2n+1(f1, f2, f3) and R1(X, Y )Z = g(Y, Z)X − g(X, Z)Y, (1) R2(X, Y )Z = g(X, ϕZ)ϕY − g(Y, ϕZ)ϕX +2g(X, ϕY )ϕZ, and (2) R3(X, Y )Z = η(X)η(Z)Y − η(Y )η(Z)X +g(X, Z)η(Y )ξ − g(Y, Z)η(X)ξ, (3) where f1 = c+3 4 , f2 = f3 = c−1 4 . Here c denotes the constant ϕ-sectional curvature. Numerous geometers, including Alegre and Carriazo [2008], De and Majhi [2015], De and Sarkar [2010], Kim [2006], Prakasha [2012], Sarkar and Akbar [2014], Sarkar and Sen [2012], Shanmukha et al. [2018], Singh [2016], W8 - Curvature tensor in generalized Sasakian-space-forms have explored the characteristics of the generalized Sasakian-space-form. Numer- ous writers have addressed the idea of local symmetry of a Riemannian manifold in various ways and to different extents. Takahashi introduced the Sasakian man- ifold’s locally ϕ-symmetry in Toshio [1977]. This is extended by De, Shaikh, and Sudipta to the notation of ϕ-symmetry in De et al. [2003], after which they introduce the notation of ϕ-recurrent Sasakian manifold. On the Kenmotsu man- ifold De et al. [2009], LP-Sasakian manifold Venkatesha [2008], W8- Curvature Tensor in the Lorentzian Sasakian manifold Rajan and Singh [2020] and (LCS)n- manifold Shaikh et al. [2008], the ϕ-recurrent condition was further studied. In Tripathi and Gupta [2012] have define the W8-curvature tensor, given by W8(X, Y )Z = R(X, Y )Z + 1 (n − 1) [S(X, Y )Z − S(Y, Z)X], where R and S are curvature tensor and Ricci tensor of the manifold respectively. A new class of almost contact Riemann manifold was presented by K. Ken- motsu [1972], sometimes referred to as a Kenmotsu manifold. Kenmotsu studied at the underlying characteristics of these manifolds local structure. Kenmotsu manifolds have a one-dimensional basis, a Kahler fibre, and are locally isomet- ric to warped product spaces. According to Kenmotsu’s research, a Kenmotsu manifold has a negative curvature of -1 if R(X,Y)Z = 0, where R is the Rieman- nian curvature tensor and R(X,Y)Z is the derivative of the tensor algebra at each point of the tangent space. Because odd dimensions hyperbolic spaces cannot ad- mit Sasakian structures, unlike odd dimensional spheres, which are well known to do so, odd dimensional hyperbolic Kenmotsu structure is permitted in spaces. Normal Kenmotsu manifolds the almost contact Riemannian manifolds. Several properties of Kenmotsu manifold have been studied by many authors like Bage- wadi et al. [2007], Blair [1976], Chaubey and Ojha [2010], DE [2008], Ingala- halli and Bagewadi [2012], Hui and Chakraborty [2017], Baishya and Chowdhury [2016], Nagaraja et al. [2018], Özgür [2006], Prakasha and Balachandra [2018], Ali Shaikh and Kumar Hui [2009], Sinha and Srivastava [1991]. These concepts served as our inspiration as we made an effort to research the characteristics of generalized Sasakian-space-form. The structure of the current paper is as follows. In section 2, we review some preliminary results. In section 3, we study ξ−W8−flat generalized Sasakian-space-forms. Section 4, deals with the ϕ−W8−semi-symmetric condition in generalized Sasakian-space-form and found to be Einstein manifold. In section 5, we discuss generalized Sasakian-space-form satisfying W8 · Q = 0 and also found to be Einstein manifold. Finally in the last Gyanvendra Pratap Singh, Rajan, Anand Kumar Mishra, Pawan Prajapati section, we discuss the generalized Sasakian-space-form satisfying W8 ·R = 0 and found to be η−Einstein Manifolds. 2 Generalized Sasakian-space-forms The Riemannian manifold M2n+1 is called an almost contact metric manifold if the following result holds Blair [1976, 2002]: ϕ2X = −X + η(X)ξ, (4) η(ξ) = 1, ϕξ = 0, η(ϕX) = 0, g(X, ξ) = η(X), (5) g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), (6) g(ϕX, Y ) = −g(X, ϕY ), g(ϕX, X) = 0, (7) (∇Xη)(Y ) = g(∇Xξ, Y ), ∀X, Y ∈ (TpM). (8) A almost contact metric manifold is said to be Sasakian if and only if Blair [1976], Sasaki [1965] (∇Xϕ)Y = g(X, Y )ξ − η(Y )X, (9) ∇Xξ = −ϕX. (10) Again we know that Alegre et al. [2004] in (2n+1)-dimensional generalized Sasakian- space -form: S(X, Y ) = (2nf1 + 3f2 − f3)g(X, Y ) − (3f2 + (2n − 1)f3)η(X)η(Y ), (11) S(ϕX, ϕY ) = S(X, Y ) + 2n(f1 − f3)η(X)η(Y ), (12) QX = (2nf1 + 3f2 − f3)X − (3f2 + (2n − 1)f3)η(X)ξ, (13) r = 2n(2n + 1)f1 + 6nf2 − 4nf3, (14) R(X, Y )ξ = (f1 − f3){η(Y )X − η(X)Y }, (15) R(ξ, X)Y = (f1 − f3){g(X, Y )ξ − η(Y )X}, (16) η(R(X, Y )Z) = (f1 − f3){g(Y, Z)η(X) − g(X, Z)η(Y )}, (17) S(X, ξ) = 2n(f1 − f3)η(X), (18) Qξ = 2n(f1 − f3)ξ, (19) for any vector fields X,Y,Z where R,S,Q and r are the Riemannian curvature ten- sor, Ricci tensor, Ricci operator g(QX, Y ) = S(X, Y ) and scalar curvature tensor W8 - Curvature tensor in generalized Sasakian-space-forms of generalized Sasakian-space-forms in that order. 3 ξ − W8−flat generalized Sasakian-space-form In this section, we study ξ − W8−flat in generalized Sasakian-space-form: Definition 3.1. A generalized Sasakian-space-form is said to be ξ − W8−flat if W8(X, Y )ξ = 0, (20) for any vector fields X, Y on M. W8-curvature tensor Tripathi and Gupta [2012] is defined as W8(X, Y )Z = R(X, Y )Z + 1 (n − 1) [S(X, Y )Z − S(Y, Z)X], (21) where R and S are curvature tensor and Ricci tensor of the manifold respectively. Replacing Z by ξ in (21), we get W8(X, Y )ξ = R(X, Y )ξ + 1 (n − 1) [S(X, Y )ξ − S(Y, ξ)X]. (22) By using (20) in (22), we get R(X, Y )ξ + 1 (n − 1) [S(X, Y )ξ − S(Y, ξ)X] = 0. (23) By virtue of (15), (18) in (23) and on simplification, we obtained (f1−f3){η(Y )X−η(X)Y }+ 1 (n − 1) [S(X, Y )ξ−2n(f1−f3)η(Y )X] = 0. (24) By taking inner product with ξ in (24) and on simplification, we have S(X, Y ) = 2n(f1 − f3)η(Y )η(X). (25) Hence above discussion, we state the following theorem: Theorem 3.1. If a generalized Sasakian-space-forms satisfying ξ − W8−flat con- dition then the generalized Sasakian-space-form is a special type of η−Einstein manifolds. Gyanvendra Pratap Singh, Rajan, Anand Kumar Mishra, Pawan Prajapati 4 ϕ − W8−semi-symmetric condition in generalized Sasakian-space-form In this section, we study ϕ − W8−semi-symmetric condition in generalized Sasakian-space-form: Definition 4.1. A generalized Sasakian-space-form is said to be ϕ − W8−semi- symmetric if W8(X, Y ) · ϕ = 0, (26) for any vector fields X,Y on M. Now, (26) turns into (W8(X, Y ) · ϕ)Z = W8(X, Y )ϕZ − ϕW8(X, Y )Z = 0. (27) From equation (21), we get W8(X, Y )Z = R(X, Y )Z + 1 (n − 1) [S(X, Y )Z − S(Y, Z)X]. (28) Replace Z by ϕZ in (28), we obtain W8(X, Y )ϕZ = R(X, Y )ϕZ + 1 (n − 1) [S(X, Y )ϕZ − S(Y, ϕZ)X]. (29) Making use of (28) and (29) in (27) and on simplification, we get R(X, Y )ϕZ − ϕR(X, Y )Z + 1 (n − 1) [S(Y, Z)ϕX − S(Y, ϕZ)X] = 0. (30) Putting X = ξ in (30) and by virtue of (16) and on simplification, we obtain (f1 − f3)g(Y, ϕZ)ξ − 1 (n − 1) S(Y, ϕZ)ξ = 0. (31) Replace ϕZ by Z in (31) and on simplification, we get S(Y, Z)ξ = (n − 1)(f1 − f3)g(Y, Z)ξ. (32) By taking inner product with ξ in (32), we get S(Y, Z) = (n − 1)(f1 − f3)g(Y, Z). (33) Hence, we state the following theorem: W8 - Curvature tensor in generalized Sasakian-space-forms Theorem 4.1. If a generalized Sasakian-space-form satisfying ϕ − W8−semi- symmetric condition then the generalized Sasakian-space-form is an Einstein man- ifolds. 5 Generalized Sasakian-space-form satisfying W8·Q = 0 In this section, we study generalized Sasakian-space-form satisfying W8 · Q = 0. Then we have W8(X, Y )QZ − Q(W8(X, Y )Z) = 0. (34) Putting Y = ξ in (34), we obtain W8(X, ξ)QZ − Q(W8(X, ξ)Z) = 0. (35) By virtue of (21) in (35), we get R(X, ξ)QZ + 1 (n − 1) [S(X, ξ)QZ − S(ξ, QZ)X] − Q{R(X, ξ)Z + 1 (n − 1) [S(X, ξ)Z − S(ξ, Z)X]} = 0. (36) By using (16), (18) in (36), we obtain −(f1 − f3)[g(X, QZ)ξ − η(QZ)X] + 1 (n − 1) [2n(f1 − f3)η(X)QZ − 2n(f1 − f3)η(QZ)X] −Q[−(f1 − f3)g(X, Z)ξ − η(Z)X] + 1 (n − 1) [2n(f1 − f3)η(X)Z − 2n(f1 − f3)η(Z)X] = 0, (37) −(f1 − f3)S(X, Z)ξ + (f1 − f3)Qη(Z)X + 2n (n − 1) (f1 − f3)η(X)QZ − 2n (n − 1) (f1 − f3)Qη(Z)X +(f1 − f3)g(X, Z)Qξ − (f1 − f3)η(Z)QX − 2n (n − 1) (f1 − f3)η(X)QZ + 2n (n − 1) (f1 − f3)Qη(Z)X = 0. (38) Gyanvendra Pratap Singh, Rajan, Anand Kumar Mishra, Pawan Prajapati Using (19) and simplifying (38), we have S(X, Z)ξ = 2n(f1 − f3)g(X, Z)ξ. (39) Taking inner product with ξ in (39) and on simplifaction, we have S(X, Z) = 2n(f1 − f3)g(X, Z). (40) Hence, we state the following theorem: Theorem 5.1. A generalized Sasakian-space-form satisfying W8 · Q = 0 is an Einstein manifolds. 6 Generalized Sasakian-space-form satisfying W8 · R = 0 In this section, we study generalized Sasakian-space-form satisfying W8 · R = 0. Then we have W8(ξ, U)R(X, Y )Z − R(W8(ξ, U)X, Y )Z −R(X, W8(ξ, U)Y )Z − R(X, Y )W8(ξ, U)Z = 0. (41) Putting Z = ξ in (41), we have W8(ξ, U)R(X, Y )ξ − R(W8(ξ, U)X, Y )ξ −R(X, W8(ξ, U)Y )ξ − R(X, Y )W8(ξ, U)ξ = 0. (42) By using (15) in (42) and on simplification, we get (f1 − f3)η(W8(ξ, U)X)Y − (f1 − f3)η(W8(ξ, U)Y )X −R(X, Y )W8(ξ, U)ξ = 0. (43) By using (21) in (43), we get (f1 − f3)η[R(ξ, U)X + 1 (n − 1) {S(ξ, U)X − S(U, X)ξ}]Y −(f1 − f3)η[R(ξ, U)Y + 1 (n − 1) {S(ξ, U)Y − S(U, Y )ξ}]X −R(X, Y )[R(ξ, U)ξ + 1 (n − 1) {S(ξ, U)ξ − S(U, ξ)ξ}] = 0. (44) W8 - Curvature tensor in generalized Sasakian-space-forms By using (16), (17), (18) in (44) and on simplification, we get (f1 − f3){g(U, X)Y − g(U, Y )X} + 2n (n − 1) (f1 − f3)η(U)η(X)Y − 2n (n − 1) (f1 − f3)η(U)η(Y )X + 1 (n − 1) {S(U, Y )X − S(U, X)Y } +R(X, Y )U = 0. (45) Putting Y = ξ in (45), we get (f1 − f3){g(U, X)ξ − g(U, ξ)X} + 2n (n − 1) (f1 − f3)η(U)η(X)ξ − 2n (n − 1) (f1 − f3)η(U)η(ξ)X + 1 (n − 1) {S(U, ξ)X − S(U, X)ξ} +R(X, ξ)U = 0. (46) By using (5), (16), (18) in (46) and on simplification, we get S(X, U)ξ = 2n(f1 − f3)η(U)η(X)ξ. (47) By taking inner product with ξ in (47), we have S(X, U) = 2n(f1 − f3)η(U)η(X). (48) Hence, we state the following theorem: Theorem 6.1. If a generalized Sasakian-space-form satisfies W8 · R = 0, then the manifold is a special type of η− Einstein manifolds. 7 Conclusions In this paper, we proposed the notion of the W8−curvature tensor in gener- alized Sasakian-space-forms drawing inspiration from the generalized Sasakian- space-form and the W8−curvature tensor. Several definitions are provided to support this new mechanism. The concept of generalized Sasakian-space-forms has been extensively studied by various authors,including Alegre et al. [2004], Prakasha [2012], Sarkar and Akbar [2014] and Shanmukha et al. [2018]. Their research has shed light on the properties and characteristics of these space-forms. Gyanvendra Pratap Singh, Rajan, Anand Kumar Mishra, Pawan Prajapati The main objective of this study is to investigate the curvature properties of the generalized Sasakian-space-form. The analysis begins by examining the ξ − W8− flat generalized Sasakian-space-forms which are revealed to be a spe- cific type of Einstein manifolds. Furthermore, the paper explores the ϕ − W8− semi-symmetric condition in the generalized Sasakian-space-form establishing it as an Einstein manifold. Additionally, the generalized Sasakian-space-form satis- fying W8 · Q = 0 is discussed demonstrating its status as an Einstein manifold. Lastly, the paper presents the discovery of generalized Sasakian-space- forms that satisfy W8 · R = 0 and are identified as η− Einstein manifolds. Statements and Declarations Funding: This work is supported by Council of Scientific and Industrial Research (CSIR), India, under Senior Research Fellowship with File.No. 09/057(0226)/2019- EMR-I. Informed Consent Statement: Not Applicable. Data Availability Statement: Data from the previous studies have been used and they are cited at the relevant places according as the reference list of the paper. Conflict of Interest: The authors declare no conflict of interest. Acknowledgements The authors are very greatful to Prof. S.K. Srivastava and Dr. Vivek Ku- mar Sharma, Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University Gorakhpur (U.P) INDIA and Prof. Ljubica S. Velimiroric, Department of Mathematics, University of NIS, Serbia for their continuous sup- port and suggestions to improve the quality and presentation of the paper. References P. Alegre and A. Carriazo. Structures on generalized sasakian-space-forms. 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