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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 317253, 10 pages
http://dx.doi.org/10.5402/2012/317253
Research Article

On πœ™-Recurrent Para-Sasakian Manifold Admitting Quarter-Symmetric Metric Connection

Department of Mathematics, Kuvempu University, Shankaraghatta, Shimoga 577 451, India

Received 3 November 2011; Accepted 6 December 2011

Academic Editors: T.Β Friedrich, M.Β Korkmaz, O.Β Mokhov, and R.Β VΓ‘zquez-Lorenzo

Copyright Β© 2012 K. T. Pradeep Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtained the relation between the Riemannian connection and the quarter-symmetric metric connection on a para-Sasakian manifold. Further, we study πœ™-recurrent and concircular πœ™-recurrent para-Sasakian manifolds with respect to quarter-symmetric metric connection.

1. Introduction

The idea of metric connection with torsion in a Riemannian manifold was introduced by Hayden [1]. Further, some properties of semisymmetric metric connection have been studied by Yano [2]. In [3], Golab defined and studied quarter-symmetric connection on a differentiable manifold with affine connection, which generalizes the idea of semisymmetric connection. Various properties of quarter-symmetric metric connection have been studied by many geometers like Rastogi [4, 5], Mishra and Pandey [6], Yano and Imai [7], De et al. [8, 9], Pradeep Kumar et al. [10], and many others.

The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, Takahashi [11] introduced the notion of local πœ™-symmetry on a Sasakian manifold. Generalizing the notion of πœ™-symmetry, the authors De et al. [12] introduced the notion of πœ™-recurrent Sasakian manifolds.

A linear connection ξ‚βˆ‡ on an 𝑛-dimensional differentiable manifold is said to be a quarter-symmetric connection [3] if its torsion tensor 𝑇 is of the form ξ‚βˆ‡π‘‡(𝑋,π‘Œ)=π‘‹ξ‚βˆ‡π‘Œβˆ’π‘Œ[]π‘‹βˆ’π‘‹,π‘Œ=πœ‚(π‘Œ)πœ™π‘‹βˆ’πœ‚(𝑋)πœ™π‘Œ,(1.1) where πœ‚ is a 1-form and πœ™ is a tensor of type (1,1). In particular, if we replace πœ™π‘‹ by 𝑋 and πœ™π‘Œ by π‘Œ, then the quarter-symmetric connection reduces to the semisymmetric connection [13]. Thus, the notion of quarter-symmetric connection generalizes the idea of the semisymmetric connection. And if quarter-symmetric linear connection ξ‚βˆ‡ satisfies the condition ξ‚€ξ‚βˆ‡π‘‹π‘”ξ‚(π‘Œ,𝑍)=0,(1.2) for all 𝑋,π‘Œ,π‘βˆˆπ’³(𝑀), where 𝒳(𝑀) is the Lie algebra of vector fields on the manifold 𝑀, then ξ‚βˆ‡ is said to be a quarter-symmetric metric connection.

2. Preliminaries

An 𝑛-dimensional differentiable manifold 𝑀 is called an almost paracontact manifold if it admits an almost paracontact structure (πœ™,πœ‰,πœ‚) consisting of a (1,1) tensor field πœ™, a vector field πœ‰, and a 1-form πœ‚ satisfying πœ™2𝑋=π‘‹βˆ’πœ‚(𝑋)πœ‰,(2.1)πœ‚(πœ‰)=1,πœ™βˆ˜πœ‰=0,πœ‚βˆ˜πœ™=0.(2.2)

If 𝑔 is a compatible Riemannian metric with (πœ™,πœ‰,πœ‚), that is, 𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ),𝑔(𝑋,πœ‰)=πœ‚(𝑋)(2.3)𝑔(𝑋,πœ™π‘Œ)=𝑔(πœ™π‘‹,π‘Œ),(2.4) for all vector fields 𝑋 and π‘Œ on 𝑀, then 𝑀 becomes a almost paracontact Riemannian manifold equipped with an almost paracontact Riemannian structure (πœ™,πœ‰,πœ‚,𝑔).

An almost paracontact Riemannian manifold is called a para-Sasakian manifold if it satisfies ξ€·βˆ‡π‘‹πœ™ξ€Έπ‘Œ=βˆ’π‘”(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋+2πœ‚(𝑋)πœ‚(π‘Œ)πœ‰,(2.5) where βˆ‡ denotes the operator of covariant differentiation. From the above equation it follows that βˆ‡π‘‹ξ€·βˆ‡πœ‰=πœ™π‘‹,π‘‹πœ‚ξ€Έξ€·βˆ‡π‘Œ=𝑔(𝑋,πœ™π‘Œ)=π‘Œπœ‚ξ€Έπ‘‹.(2.6)

In an 𝑛-dimensional para-Sasakian manifold 𝑀, the following relations hold [14, 15]: πœ‚(𝑅(𝑋,π‘Œ)𝑍)=𝑔(𝑋,𝑍)πœ‚(π‘Œ)βˆ’π‘”(π‘Œ,𝑍)πœ‚(𝑋),(2.7)𝑅(𝑋,π‘Œ)πœ‰=πœ‚(𝑋)π‘Œβˆ’πœ‚(π‘Œ)𝑋,(2.8)𝑆(𝑋,πœ‰)=βˆ’(π‘›βˆ’1)πœ‚(𝑋),(2.9)𝑆(πœ™π‘‹,πœ™π‘Œ)=𝑆(𝑋,π‘Œ)+(π‘›βˆ’1)πœ‚(𝑋)πœ‚(π‘Œ),(2.10) for any vector fields 𝑋,π‘Œ, and 𝑍, where 𝑅 and 𝑆 are the Riemannian curvature tensor and the Ricci tensor of 𝑀, respectively.

A para-Sasakian manifold 𝑀 is said to be πœ‚-Einstein if its Ricci tensor 𝑆 is of the form 𝑆(𝑋,π‘Œ)=π‘Žπ‘”(𝑋,π‘Œ)+π‘πœ‚(𝑋)πœ‚(π‘Œ),(2.11) for any vector fields 𝑋 and π‘Œ, where π‘Ž and 𝑏 are some functions on 𝑀.

Definition 2.1. A para-Sasakian manifold is said to be locally πœ™-symmetric if πœ™2βˆ‡ξ€·ξ€·π‘Šπ‘…ξ€Έξ€Έ(𝑋,π‘Œ)𝑍=0,(2.12) for all vector fields 𝑋,π‘Œ,𝑍,π‘Š orthogonal to πœ‰. This notion was introduced for Sasakian manifold by Takahashi [11].

Definition 2.2. A para-Sasakian manifold is said to be locally concircular πœ™-symmetric if πœ™2βˆ‡ξ‚€ξ‚€π‘ŠπΆξ‚ξ‚(𝑋,π‘Œ)𝑍=0,(2.13) for all vector fields 𝑋,π‘Œ,𝑍,π‘Š orthogonal to πœ‰. Where the concircular curvature tensor 𝐢 is given by [16] πΆπ‘Ÿ(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)π‘βˆ’[𝑔],𝑛(π‘›βˆ’1)(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(2.14) where 𝑅 is the Riemannian curvature tensor and π‘Ÿ is the scalar curvature.

Definition 2.3. A para-Sasakian manifold is said to be πœ™-recurrent if there exists a nonzero 1-form 𝐴 such that πœ™2βˆ‡ξ€·ξ€·π‘Šπ‘…ξ€Έξ€Έ(𝑋,π‘Œ)𝑍=𝐴(π‘Š)𝑅(𝑋,π‘Œ)𝑍,(2.15) where 𝐴 is a 1-form and it is defined by 𝐴(π‘Š)=𝑔(π‘Š,𝜌),(2.16) and 𝜌 is a vector field associated with the 1-form 𝐴.

3. Quarter-Symmetric Metric Connection

Let ξ‚βˆ‡ be a linear connection and βˆ‡ a Riemannian connection of an almost contact metric manifold 𝑀 such that ξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+π‘ˆ(𝑋,π‘Œ),(3.1) where π‘ˆ is a tensor of type (1,1). For ξ‚βˆ‡ to be a quarter-symmetric metric connection in 𝑀, then we have [3] 1π‘ˆ(𝑋,π‘Œ)=2𝑇(𝑋,π‘Œ)+π‘‡ξ…ž(𝑋,π‘Œ)+π‘‡ξ…žξ€»,(π‘Œ,𝑋)(3.2)π‘”ξ€·π‘‡ξ…žξ€Έ(𝑋,π‘Œ),𝑍=𝑔(𝑇(𝑍,𝑋),π‘Œ).(3.3) From (1.1) and (3.3), we get π‘‡ξ…ž(𝑋,π‘Œ)=πœ‚(𝑋)πœ™π‘Œβˆ’π‘”(πœ™π‘‹,π‘Œ)πœ‰.(3.4) Using (1.1) and (3.4) in (3.2), we obtain π‘ˆ(𝑋,π‘Œ)=πœ‚(π‘Œ)πœ™π‘‹βˆ’π‘”(πœ™π‘‹,π‘Œ)πœ‰.(3.5) Thus a quarter-symmetric metric connection ξ‚βˆ‡ in a para-Sasakian manifold is given by ξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+πœ‚(π‘Œ)πœ™π‘‹βˆ’π‘”(πœ™π‘‹,π‘Œ)πœ‰.(3.6) Hence (3.6) is the relation between Riemannian connection and the quarter-symmetric metric connection on a para-Sasakian manifold.

A relation between the curvature tensor of 𝑀 with respect to the quarter-symmetric metric connection ξ‚βˆ‡ and the Riemannian connection βˆ‡ is given by 𝑅[πœ‚]βˆ’[](𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)𝑍+3𝑔(πœ™π‘‹,𝑍)πœ™π‘Œβˆ’3𝑔(πœ™π‘Œ,𝑍)πœ™π‘‹+πœ‚(𝑍)(𝑋)π‘Œβˆ’πœ‚(π‘Œ)π‘‹πœ‚(𝑋)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(𝑋,𝑍)πœ‰,(3.7) where 𝑅 and 𝑅 denote the Riemannian curvatures of the connections ξ‚βˆ‡ and βˆ‡, respectively. From (3.7), it follows that 𝑆(π‘Œ,𝑍)=𝑆(π‘Œ,𝑍)+2𝑔(π‘Œ,𝑍)βˆ’(𝑛+1)πœ‚(π‘Œ)πœ‚(𝑍),(3.8) where 𝑆 and 𝑆 are the Ricci tensors of the connections ξ‚βˆ‡ and βˆ‡, respectively.

Contracting (3.8), we get Μƒπ‘Ÿ=π‘Ÿ+(π‘›βˆ’1),(3.9) where Μƒπ‘Ÿ and π‘Ÿ are the scalar curvatures of the connections ξ‚βˆ‡ and βˆ‡, respectively.

4. πœ™-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A para-Sasakian manifold is called πœ™-recurrent with respect to the quarter-symmetric metric connection if its curvature tensor 𝑅 satisfies the condition πœ™2ξ‚βˆ‡ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ‚ξ‚(𝑋,π‘Œ)𝑍=𝐴(π‘Š)𝑅(𝑋,π‘Œ)𝑍.(4.1) By virtue of (2.1) and (4.1), we have ξ‚€ξ‚βˆ‡π‘Šξ‚π‘…ξ‚ξ‚βˆ‡(𝑋,π‘Œ)π‘βˆ’πœ‚ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ‚ξ‚(𝑋,π‘Œ)π‘πœ‰=𝐴(π‘Š)𝑅(𝑋,π‘Œ)𝑍.(4.2) From which, it follows that π‘”ξ‚βˆ‡ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ‚ξ‚βˆ‡(𝑋,π‘Œ)𝑍,π‘ˆβˆ’πœ‚ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ‚ξ‚€ξ‚ξ‚.(𝑋,π‘Œ)𝑍𝑔(πœ‰,π‘ˆ)=𝐴(π‘Š)𝑔𝑅(𝑋,π‘Œ)𝑍,π‘ˆ(4.3) Let {𝑒𝑖},𝑖=1,2,…,𝑛 be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=π‘ˆ=𝑒𝑖 in (4.3) and taking summation over 𝑖,  1≀𝑖≀𝑛, we get ξ‚€ξ‚βˆ‡π‘Šξ‚π‘†ξ‚(π‘Œ,𝑍)βˆ’π‘›ξ“π‘–=1πœ‚ξ‚βˆ‡ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ€·π‘’π‘–ξ€Έπ‘ξ‚πœ‚ξ€·π‘’,π‘Œπ‘–ξ€Έξ‚=𝐴(π‘Š)𝑆(π‘Œ,𝑍).(4.4) The second term of (4.4) by putting 𝑍=πœ‰ takes the form π‘”ξ‚βˆ‡ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ€·π‘’π‘–ξ€Έξ‚ξ‚€ξ‚βˆ‡,π‘Œπœ‰,πœ‰=π‘”π‘Šξ‚π‘…ξ€·π‘’π‘–ξ€Έξ‚ξ‚€ξ‚π‘…ξ‚€ξ‚βˆ‡,π‘Œπœ‰,πœ‰βˆ’π‘”π‘Šπ‘’π‘–ξ‚ξ‚ξ‚€ξ‚π‘…ξ‚€π‘’,π‘Œπœ‰,πœ‰βˆ’π‘”π‘–,ξ‚βˆ‡π‘Šπ‘Œξ‚ξ‚ξ‚€ξ‚π‘…ξ€·π‘’πœ‰,πœ‰βˆ’π‘”π‘–ξ€Έξ‚βˆ‡,π‘Œπ‘Šξ‚.πœ‰,πœ‰(4.5) On simplification we obtain π‘”ξ‚βˆ‡ξ‚€ξ‚€π‘Šξ‚π‘…ξ‚ξ€·π‘’π‘–ξ€Έξ‚,π‘Œπ‘,πœ‰=0.(4.6) Therefore (4.4) can be written in the form ξ‚€ξ‚βˆ‡π‘Šξ‚π‘†ξ‚ξ‚(π‘Œ,𝑍)=𝐴(π‘Š)𝑆(π‘Œ,𝑍).(4.7) Replacing 𝑍 by πœ‰ in the above relation, then using (3.8) and (2.9), we haveξ‚€ξ‚βˆ‡π‘Šξ‚π‘†ξ‚(π‘Œ,πœ‰)=βˆ’2(π‘›βˆ’1)𝐴(π‘Š)πœ‚(π‘Œ).(4.8) We know that ξ‚€ξ‚βˆ‡π‘Šξ‚π‘†ξ‚ξ‚βˆ‡(π‘Œ,πœ‰)=π‘Šξ‚ξ‚π‘†ξ‚€ξ‚βˆ‡π‘†(π‘Œ,πœ‰)βˆ’π‘Šξ‚βˆ’ξ‚π‘†ξ‚€ξ‚βˆ‡π‘Œ,πœ‰π‘Œ,π‘Šπœ‰ξ‚.(4.9) Using (3.8), (2.6) and (2.9) in the above relation, we get ξ‚€ξ‚βˆ‡π‘Šξ‚π‘†ξ‚(π‘Œ,πœ‰)=βˆ’4(π‘›βˆ’1)𝑔(π‘Œ,πœ™π‘Š)βˆ’2𝑆(π‘Œ,πœ™π‘Š)+4𝑔(π‘Œ,πœ™π‘Š).(4.10) In view of (4.8) and (4.10), we obtain βˆ’4(π‘›βˆ’1)𝑔(π‘Œ,πœ™π‘Š)βˆ’2𝑆(π‘Œ,πœ™π‘Š)+4𝑔(π‘Œ,πœ™π‘Š)=βˆ’2(π‘›βˆ’1)𝐴(π‘Š)πœ‚(π‘Œ).(4.11) Replacing π‘Œ by πœ™π‘Œ in (4.11) and then using (2.3) and (2.10), we have 𝑆(π‘Œ,π‘Š)=βˆ’2(π‘›βˆ’2)𝑔(π‘Œ,π‘Š)+(π‘›βˆ’3)πœ‚(π‘Œ)πœ‚(π‘Š).(4.12) Hence, we can state the following.

Theorem 4.1. If para-Sasakian manifold is πœ™-recurrent with respect to quarter-symmetric metric connection then it is an πœ‚-Einstein manifold with respect to Riemannian connection.

5. Concircular πœ™-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A concircular πœ™-recurrent para-Sasakian manifold with respect to the quarter-symmetric metric connection is defined by πœ™2ξ‚βˆ‡ξ‚΅ξ‚΅π‘Šξ‚πΆξ‚Άξ‚Άξ‚(𝑋,π‘Œ)𝑍=𝐴(π‘Š)𝐢(𝑋,π‘Œ)𝑍,(5.1) where 𝐢 is a concircular curvature tensor with respect to the quarter-symmetric metric connection given by 𝐢(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)π‘βˆ’Μƒπ‘Ÿ[].𝑛(π‘›βˆ’1)𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(5.2) By virtue of (2.1) and (5.1), we have ξ‚΅ξ‚βˆ‡π‘Šξ‚πΆξ‚Άξ‚βˆ‡(𝑋,π‘Œ)π‘βˆ’πœ‚ξ‚΅ξ‚΅π‘Šξ‚πΆξ‚Άξ‚Άξ‚(𝑋,π‘Œ)π‘πœ‰=𝐴(π‘Š)𝐢(𝑋,π‘Œ)𝑍,(5.3) from which it follows that π‘”ξ‚βˆ‡ξ‚΅ξ‚΅π‘Šξ‚πΆξ‚Άξ‚Άξ‚βˆ‡(𝑋,π‘Œ)𝑍,π‘ˆβˆ’πœ‚ξ‚΅ξ‚΅π‘Šξ‚πΆξ‚Άξ‚Άξ‚΅ξ‚(𝑋,π‘Œ)𝑍𝑔(πœ‰,π‘ˆ)=𝐴(π‘Š)𝑔,𝐢(𝑋,π‘Œ)𝑍,π‘ˆ(5.4) where ξ‚΅ξ‚βˆ‡π‘Šξ‚πΆξ‚Άβˆ‡(𝑋,π‘Œ)𝑍=ξ€·ξ€·π‘Šπ‘…ξ€Έξ€Έ[]πœ‰[πœ‚][][][πœ‚]π‘Š[][][][πœ‚]πœ‰[][][]+[]πœ‰βˆ’[𝑔][πœ‚]βˆ’βˆ‡(𝑋,π‘Œ)𝑍+6𝑔(πœ™π‘Œ,𝑍)𝑔(π‘Š,𝑋)βˆ’π‘”(πœ™π‘‹,𝑍)𝑔(π‘Š,π‘Œ)+6(π‘Œ)𝑔(π‘Š,𝑍)+πœ‚(𝑍)𝑔(π‘Š,π‘Œ)πœ™π‘‹βˆ’6πœ‚(𝑋)𝑔(π‘Š,𝑍)+πœ‚(𝑍)𝑔(π‘Š,𝑋)πœ™π‘Œ+2πœ‚(π‘Œ)𝑔(𝑋,𝑍)βˆ’πœ‚(𝑋)𝑔(π‘Œ,𝑍)πœ™π‘Š+6(𝑋)𝑔(πœ™π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(πœ™π‘‹,𝑍)+12πœ‚(π‘Š)πœ‚(𝑍)πœ‚(𝑋)πœ™π‘Œβˆ’πœ‚(π‘Œ)πœ™π‘‹+πœ‚(𝑍)𝑔(π‘Š,π‘Œ)π‘‹βˆ’π‘”(π‘Š,𝑋)π‘Œ+2πœ‚(π‘Š)πœ‚(𝑍)πœ‚(𝑋)π‘Œβˆ’πœ‚(π‘Œ)𝑋+12πœ‚(π‘Š)(π‘Œ)𝑔(πœ™π‘‹,𝑍)βˆ’πœ‚(𝑋)𝑔(πœ™π‘Œ,𝑍)+πœ‚(π‘Š)πœ‚(π‘Œ)𝑔(𝑋,𝑍)βˆ’πœ‚(𝑋)𝑔(π‘Œ,𝑍)πœ‰+πœ‚(𝑍)𝑔(πœ™π‘Š,𝑋)π‘Œβˆ’π‘”(πœ™π‘Š,π‘Œ)𝑋+𝑔(π‘Š,𝑍)πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œπ‘”(π‘Š,𝑋)𝑔(π‘Œ,𝑍)βˆ’π‘”(π‘Š,π‘Œ)𝑔(𝑋,𝑍)(πœ™π‘Š,𝑋)𝑔(π‘Œ,𝑍)βˆ’π‘”(πœ™π‘Š,π‘Œ)𝑔(𝑋,𝑍)πœ‰+𝑔(πœ™π‘Š,𝑍)(𝑋)π‘Œβˆ’πœ‚(π‘Œ)π‘‹π‘Šπ‘Ÿ[].𝑛(π‘›βˆ’1)𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(5.5) Let {𝑒𝑖},𝑖=1,2,…,𝑛 be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=π‘ˆ=𝑒𝑖 in (5.4) and taking summation over 𝑖,1≀𝑖≀𝑛, we get ξ€·βˆ‡π‘Šπ‘†ξ€Έβˆ‡(π‘Œ,𝑍)=π‘Šπ‘Ÿπ‘›βˆ’βˆ‡π‘”(π‘Œ,𝑍)+(𝑛+4)πœ‚(𝑍)𝑔(πœ™π‘Š,π‘Œ)+(𝑛+3)πœ‚(π‘Œ)𝑔(πœ™π‘Š,𝑍)+(2π‘›βˆ’3)πœ‚(π‘Š)πœ‚(π‘Œ)πœ‚(𝑍)βˆ’(π‘›βˆ’1)πœ‚(π‘Œ)𝑔(π‘Š,𝑍)π‘Šπ‘Ÿ[]𝑛(π‘›βˆ’1)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)πœ‚(𝑍)+𝐴(π‘Š)𝑆(π‘Œ,𝑍)βˆ’π΄(π‘Š)(𝑛+1)πœ‚(π‘Œ)πœ‚(𝑍)+π‘Ÿβˆ’(𝑛+1)𝑛.𝑔(π‘Œ,𝑍)(5.6) Replacing 𝑍 by πœ‰ in (5.6) and using (2.9), we have ξ€·βˆ‡π‘Šπ‘†ξ€Έβˆ‡(π‘Œ,πœ‰)=π‘Šπ‘Ÿπ‘›ξ‚Έπœ‚(π‘Œ)+(𝑛+4)𝑔(πœ™π‘Š,π‘Œ)+(π‘›βˆ’2)πœ‚(π‘Š)πœ‚(π‘Œ)βˆ’π΄(π‘Š)πœ‚(π‘Œ)2𝑛+π‘Ÿβˆ’(𝑛+1)𝑛.(5.7) We know that ξ€·βˆ‡π‘Šπ‘†ξ€Έ(π‘Œ,πœ‰)=βˆ‡π‘Šπ‘†ξ€·βˆ‡(π‘Œ,πœ‰)βˆ’π‘†π‘Šξ€Έξ€·π‘Œ,πœ‰βˆ’π‘†π‘Œ,βˆ‡π‘Šπœ‰ξ€Έ.(5.8) Using (2.6) and (2.9) in the above relation, it follows that ξ€·βˆ‡π‘Šπ‘†ξ€Έ[𝑔](π‘Œ,πœ‰)=βˆ’(π‘›βˆ’1)(πœ™π‘Š,π‘Œ)βˆ’π‘†(π‘Œ,πœ™π‘Š).(5.9) In view of (5.7) and (5.9), we obtain βˆ‡π‘†(π‘Œ,πœ™π‘Š)=βˆ’(π‘›βˆ’1)𝑔(πœ™π‘Š,π‘Œ)βˆ’π‘Šπ‘Ÿπ‘›ξ‚Έπœ‚(π‘Œ)βˆ’(𝑛+4)𝑔(πœ™π‘Š,π‘Œ)βˆ’(π‘›βˆ’2)πœ‚(π‘Š)πœ‚(π‘Œ)+𝐴(π‘Š)πœ‚(π‘Œ)2𝑛+π‘Ÿβˆ’(𝑛+1)𝑛.(5.10) Replacing π‘Œ by πœ™π‘Œ in (5.10) and then using (2.3) and (2.10), we obtain 𝑆(π‘Œ,π‘Š)=βˆ’(2𝑛+3)𝑔(π‘Š,π‘Œ)+(𝑛+4)πœ‚(π‘Š)πœ‚(π‘Œ).(5.11) This leads to the following theorem.

Theorem 5.1. If para-Sasakian manifold is concircular πœ™-recurrent with respect to quarter-symmetric metric connection then it is an πœ‚-Einstein manifold with respect to Riemannian connection.

Now from (5.3), we have ξ‚΅ξ‚βˆ‡π‘Šξ‚πΆξ‚Άξ‚βˆ‡(𝑋,π‘Œ)𝑍=πœ‚ξ‚΅ξ‚΅π‘Šξ‚πΆξ‚Άξ‚Άξ‚(𝑋,π‘Œ)π‘πœ‰+𝐴(π‘Š)𝐢(𝑋,π‘Œ)𝑍.(5.12) This gives βˆ‡ξ€·ξ€·π‘Šπ‘…ξ€Έξ€Έβˆ‡(𝑋,π‘Œ)𝑍=πœ‚ξ€·ξ€·π‘Šπ‘…ξ€Έξ€Έ[πœ‚][][][][][][][][]πœ‰[][][][]πœ‰+βˆ‡(𝑋,π‘Œ)π‘πœ‰+6(π‘Œ)𝑔(π‘Š,𝑍)βˆ’πœ‚(𝑍)𝑔(π‘Š,π‘Œ)πœ™π‘‹+6πœ‚(𝑋)𝑔(π‘Š,𝑍)+πœ‚(𝑍)𝑔(π‘Š,𝑋)πœ™π‘Œ+2πœ‚(𝑋)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(𝑋,𝑍)πœ™π‘Šβˆ’6πœ‚(𝑋)𝑔(πœ™π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(πœ™π‘‹,𝑍)π‘Šβˆ’2πœ‚(π‘Š)πœ‚(𝑍)πœ‚(𝑋)π‘Œβˆ’πœ‚(π‘Œ)𝑋+12πœ‚(π‘Š)πœ‚(𝑍)πœ‚(π‘Œ)πœ™π‘‹βˆ’πœ‚(𝑋)πœ™π‘Œβˆ’πœ‚(𝑍)𝑔(π‘Š,π‘Œ)π‘‹βˆ’π‘”(π‘Š,𝑋)π‘Œ+πœ‚(𝑍)𝑔(πœ™π‘Š,π‘Œ)π‘‹βˆ’π‘”(πœ™π‘Š,𝑋)π‘Œ+πœ‚(𝑍)πœ‚(𝑋)𝑔(π‘Š,π‘Œ)βˆ’πœ‚(π‘Œ)𝑔(π‘Š,𝑋)+πœ‚(𝑍)πœ‚(π‘Œ)𝑔(πœ™π‘Š,𝑋)βˆ’πœ‚(𝑋)𝑔(πœ™π‘Š,π‘Œ)πœ‰βˆ’π‘”(π‘Š,𝑍)πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ+𝑔(πœ™π‘Š,𝑍)πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ+6πœ‚(π‘Š)πœ‚(𝑋)𝑔(πœ™π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(πœ™π‘‹,𝑍)π‘Šπ‘Ÿ[][𝑔][][]πœ‰βˆ’π‘›(π‘›βˆ’1)𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œβˆ’πœ‚(𝑋)𝑔(π‘Œ,𝑍)πœ‰+πœ‚(π‘Œ)𝑔(𝑋,𝑍)πœ‰+𝐴(π‘Š)𝑅(𝑋,π‘Œ)𝑍+3𝐴(π‘Š)(πœ™π‘‹,𝑍)πœ™π‘Œβˆ’π‘”(πœ™π‘Œ,𝑍)πœ™π‘‹+𝐴(π‘Š)πœ‚(𝑍)πœ‚(𝑋)π‘Œβˆ’πœ‚(π‘Œ)π‘‹βˆ’π΄(π‘Š)πœ‚(𝑋)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(𝑋,𝑍)π‘Ÿ+(π‘›βˆ’1)[].𝑛(π‘›βˆ’1)𝐴(π‘Š)𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(5.13) Now from (5.13) and Bianchi’s second identity, we have =𝐴(π‘Š)πœ‚(𝑅(𝑋,π‘Œ)𝑍)+𝐴(𝑋)πœ‚(𝑅(π‘Œ,π‘Š)𝑍)+𝐴(π‘Œ)πœ‚(𝑅(π‘Š,𝑋)𝑍)(𝑛+1)(π‘›βˆ’1)+π‘Ÿ[]+𝑛(π‘›βˆ’1)𝐴(π‘Š)πœ‚(𝑋)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(𝑋,𝑍)(𝑛+1)(π‘›βˆ’1)+π‘Ÿπ‘›[]+(π‘›βˆ’1)𝐴(𝑋)πœ‚(π‘Œ)𝑔(π‘Š,𝑍)βˆ’πœ‚(π‘Š)𝑔(π‘Œ,𝑍)(𝑛+1)(π‘›βˆ’1)+π‘Ÿ[].𝑛(π‘›βˆ’1)𝐴(π‘Œ)πœ‚(π‘Š)𝑔(𝑋,𝑍)βˆ’πœ‚(𝑋)𝑔(π‘Š,𝑍)(5.14) By virtue of (2.7), we obtain from (5.14) that [][𝑔][]=𝐴(π‘Š)𝑔(𝑋,𝑍)πœ‚(π‘Œ)βˆ’π‘”(π‘Œ,𝑍)πœ‚(𝑋)+𝐴(𝑋)(π‘Œ,𝑍)πœ‚(π‘Š)βˆ’π‘”(π‘Š,𝑍)πœ‚(π‘Œ)+𝐴(π‘Œ)𝑔(π‘Š,𝑍)πœ‚(𝑋)βˆ’π‘”(𝑋,𝑍)πœ‚(π‘Š)(𝑛+1)(π‘›βˆ’1)+π‘Ÿ[]+𝑛(π‘›βˆ’1)𝐴(π‘Š)πœ‚(𝑋)𝑔(π‘Œ,𝑍)βˆ’πœ‚(π‘Œ)𝑔(𝑋,𝑍)(𝑛+1)(π‘›βˆ’1)+π‘Ÿ[]+𝑛(π‘›βˆ’1)𝐴(𝑋)πœ‚(π‘Œ)𝑔(π‘Š,𝑍)βˆ’πœ‚(π‘Š)𝑔(π‘Œ,𝑍)(𝑛+1)(π‘›βˆ’1)+π‘Ÿ[].𝑛(π‘›βˆ’1)𝐴(π‘Œ)πœ‚(π‘Š)𝑔(𝑋,𝑍)βˆ’πœ‚(𝑋)𝑔(π‘Š,𝑍)(5.15) Putting π‘Œ=𝑍=𝑒𝑖 in (5.15) and taking summation over 𝑖, 1≀𝑖≀𝑛, we get 𝐴(π‘Š)πœ‚(𝑋)=𝐴(𝑋)πœ‚(π‘Š),(5.16) for all vector fields 𝑋,π‘Š. Replacing 𝑋 by πœ‰ in (5.16), we get 𝐴(π‘Š)=πœ‚(π‘Š)πœ‚(𝜌),(5.17) for any vector field π‘Š.

Hence from (5.16) and (5.17), we can state the following.

Theorem 5.2. In a concircular πœ™-recurrent para-Sasakian manifold with respect to quarter-symmetric metric connection, the characteristic vector field πœ‰ and the vector field 𝜌 associated to the 1-form 𝐴 are in codirectional and the 1-form 𝐴 is given by (5.17).

Acknowledgments

The authors express their thanks to DST (Department of Science and Technology), Government of India, for providing financial assistance under major research project (no. SR/S4/MS: 482/07). They are also thankful to the referees for valuable suggestions.

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