Abstract

We obtained the relation between the Riemannian connection and the quarter-symmetric metric connection on a para-Sasakian manifold. Further, we study ๐œ™-recurrent and concircular ๐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 ๐œ™-symmetry on a Sasakian manifold. Generalizing the notion of ๐œ™-symmetry, the authors De et al. [12] introduced the notion of ๐œ™-recurrent Sasakian manifolds.

A linear connection ๎‚โˆ‡ on an ๐‘›-dimensional differentiable manifold is said to be a quarter-symmetric connection [3] if its torsion tensor ๐‘‡ is of the form ๎‚โˆ‡๐‘‡(๐‘‹,๐‘Œ)=๐‘‹๎‚โˆ‡๐‘Œโˆ’๐‘Œ[]๐‘‹โˆ’๐‘‹,๐‘Œ=๐œ‚(๐‘Œ)๐œ™๐‘‹โˆ’๐œ‚(๐‘‹)๐œ™๐‘Œ,(1.1) where ๐œ‚ is a 1-form and ๐œ™ is a tensor of type (1,1). In particular, if we replace ๐œ™๐‘‹ by ๐‘‹ and ๐œ™๐‘Œ by ๐‘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 ๎‚โˆ‡ satisfies the condition ๎‚€๎‚โˆ‡๐‘‹๐‘”๎‚(๐‘Œ,๐‘)=0,(1.2) for all ๐‘‹,๐‘Œ,๐‘โˆˆ๐’ณ(๐‘€), where ๐’ณ(๐‘€) is the Lie algebra of vector fields on the manifold ๐‘€, then ๎‚โˆ‡ is said to be a quarter-symmetric metric connection.

2. Preliminaries

An ๐‘›-dimensional differentiable manifold ๐‘€ is called an almost paracontact manifold if it admits an almost paracontact structure (๐œ™,๐œ‰,๐œ‚) consisting of a (1,1) tensor field ๐œ™, a vector field ๐œ‰, and a 1-form ๐œ‚ satisfying ๐œ™2๐‘‹=๐‘‹โˆ’๐œ‚(๐‘‹)๐œ‰,(2.1)๐œ‚(๐œ‰)=1,๐œ™โˆ˜๐œ‰=0,๐œ‚โˆ˜๐œ™=0.(2.2)

If ๐‘” is a compatible Riemannian metric with (๐œ™,๐œ‰,๐œ‚), that is, ๐‘”(๐œ™๐‘‹,๐œ™๐‘Œ)=๐‘”(๐‘‹,๐‘Œ)โˆ’๐œ‚(๐‘‹)๐œ‚(๐‘Œ),๐‘”(๐‘‹,๐œ‰)=๐œ‚(๐‘‹)(2.3)๐‘”(๐‘‹,๐œ™๐‘Œ)=๐‘”(๐œ™๐‘‹,๐‘Œ),(2.4) for all vector fields ๐‘‹ and ๐‘Œ on ๐‘€, then ๐‘€ becomes a almost paracontact Riemannian manifold equipped with an almost paracontact Riemannian structure (๐œ™,๐œ‰,๐œ‚,๐‘”).

An almost paracontact Riemannian manifold is called a para-Sasakian manifold if it satisfies ๎€ทโˆ‡๐‘‹๐œ™๎€ธ๐‘Œ=โˆ’๐‘”(๐‘‹,๐‘Œ)๐œ‰โˆ’๐œ‚(๐‘Œ)๐‘‹+2๐œ‚(๐‘‹)๐œ‚(๐‘Œ)๐œ‰,(2.5) where โˆ‡ denotes the operator of covariant differentiation. From the above equation it follows that โˆ‡๐‘‹๎€ทโˆ‡๐œ‰=๐œ™๐‘‹,๐‘‹๐œ‚๎€ธ๎€ทโˆ‡๐‘Œ=๐‘”(๐‘‹,๐œ™๐‘Œ)=๐‘Œ๐œ‚๎€ธ๐‘‹.(2.6)

In an ๐‘›-dimensional para-Sasakian manifold ๐‘€, the following relations hold [14, 15]: ๐œ‚(๐‘…(๐‘‹,๐‘Œ)๐‘)=๐‘”(๐‘‹,๐‘)๐œ‚(๐‘Œ)โˆ’๐‘”(๐‘Œ,๐‘)๐œ‚(๐‘‹),(2.7)๐‘…(๐‘‹,๐‘Œ)๐œ‰=๐œ‚(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹,(2.8)๐‘†(๐‘‹,๐œ‰)=โˆ’(๐‘›โˆ’1)๐œ‚(๐‘‹),(2.9)๐‘†(๐œ™๐‘‹,๐œ™๐‘Œ)=๐‘†(๐‘‹,๐‘Œ)+(๐‘›โˆ’1)๐œ‚(๐‘‹)๐œ‚(๐‘Œ),(2.10) for any vector fields ๐‘‹,๐‘Œ, and ๐‘, where ๐‘… and ๐‘† are the Riemannian curvature tensor and the Ricci tensor of ๐‘€, respectively.

A para-Sasakian manifold ๐‘€ is said to be ๐œ‚-Einstein if its Ricci tensor ๐‘† is of the form ๐‘†(๐‘‹,๐‘Œ)=๐‘Ž๐‘”(๐‘‹,๐‘Œ)+๐‘๐œ‚(๐‘‹)๐œ‚(๐‘Œ),(2.11) for any vector fields ๐‘‹ and ๐‘Œ, where ๐‘Ž and ๐‘ are some functions on ๐‘€.

Definition 2.1. A para-Sasakian manifold is said to be locally ๐œ™-symmetric if ๐œ™2โˆ‡๎€ท๎€ท๐‘Š๐‘…๎€ธ๎€ธ(๐‘‹,๐‘Œ)๐‘=0,(2.12) for all vector fields ๐‘‹,๐‘Œ,๐‘,๐‘Š orthogonal to ๐œ‰. This notion was introduced for Sasakian manifold by Takahashi [11].

Definition 2.2. A para-Sasakian manifold is said to be locally concircular ๐œ™-symmetric if ๐œ™2โˆ‡๎‚€๎‚€๐‘Š๐ถ๎‚๎‚(๐‘‹,๐‘Œ)๐‘=0,(2.13) for all vector fields ๐‘‹,๐‘Œ,๐‘,๐‘Š orthogonal to ๐œ‰. Where the concircular curvature tensor ๐ถ is given by [16] ๐ถ๐‘Ÿ(๐‘‹,๐‘Œ)๐‘=๐‘…(๐‘‹,๐‘Œ)๐‘โˆ’[๐‘”],๐‘›(๐‘›โˆ’1)(๐‘Œ,๐‘)๐‘‹โˆ’๐‘”(๐‘‹,๐‘)๐‘Œ(2.14) where ๐‘… is the Riemannian curvature tensor and ๐‘Ÿ is the scalar curvature.

Definition 2.3. A para-Sasakian manifold is said to be ๐œ™-recurrent if there exists a nonzero 1-form ๐ด such that ๐œ™2โˆ‡๎€ท๎€ท๐‘Š๐‘…๎€ธ๎€ธ(๐‘‹,๐‘Œ)๐‘=๐ด(๐‘Š)๐‘…(๐‘‹,๐‘Œ)๐‘,(2.15) where ๐ด is a 1-form and it is defined by ๐ด(๐‘Š)=๐‘”(๐‘Š,๐œŒ),(2.16) and ๐œŒ is a vector field associated with the 1-form ๐ด.

3. Quarter-Symmetric Metric Connection

Let ๎‚โˆ‡ be a linear connection and โˆ‡ a Riemannian connection of an almost contact metric manifold ๐‘€ such that ๎‚โˆ‡๐‘‹๐‘Œ=โˆ‡๐‘‹๐‘Œ+๐‘ˆ(๐‘‹,๐‘Œ),(3.1) where ๐‘ˆ is a tensor of type (1,1). For ๎‚โˆ‡ to be a quarter-symmetric metric connection in ๐‘€, then we have [3] 1๐‘ˆ(๐‘‹,๐‘Œ)=2๎€บ๐‘‡(๐‘‹,๐‘Œ)+๐‘‡๎…ž(๐‘‹,๐‘Œ)+๐‘‡๎…ž๎€ป,(๐‘Œ,๐‘‹)(3.2)๐‘”๎€ท๐‘‡๎…ž๎€ธ(๐‘‹,๐‘Œ),๐‘=๐‘”(๐‘‡(๐‘,๐‘‹),๐‘Œ).(3.3) From (1.1) and (3.3), we get ๐‘‡๎…ž(๐‘‹,๐‘Œ)=๐œ‚(๐‘‹)๐œ™๐‘Œโˆ’๐‘”(๐œ™๐‘‹,๐‘Œ)๐œ‰.(3.4) Using (1.1) and (3.4) in (3.2), we obtain ๐‘ˆ(๐‘‹,๐‘Œ)=๐œ‚(๐‘Œ)๐œ™๐‘‹โˆ’๐‘”(๐œ™๐‘‹,๐‘Œ)๐œ‰.(3.5) Thus a quarter-symmetric metric connection ๎‚โˆ‡ in a para-Sasakian manifold is given by ๎‚โˆ‡๐‘‹๐‘Œ=โˆ‡๐‘‹๐‘Œ+๐œ‚(๐‘Œ)๐œ™๐‘‹โˆ’๐‘”(๐œ™๐‘‹,๐‘Œ)๐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 ๐‘€ with respect to the quarter-symmetric metric connection ๎‚โˆ‡ and the Riemannian connection โˆ‡ is given by ๎‚๐‘…[๐œ‚]โˆ’[](๐‘‹,๐‘Œ)๐‘=๐‘…(๐‘‹,๐‘Œ)๐‘+3๐‘”(๐œ™๐‘‹,๐‘)๐œ™๐‘Œโˆ’3๐‘”(๐œ™๐‘Œ,๐‘)๐œ™๐‘‹+๐œ‚(๐‘)(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)๐œ‰,(3.7) where ๎‚๐‘… and ๐‘… denote the Riemannian curvatures of the connections ๎‚โˆ‡ and โˆ‡, respectively. From (3.7), it follows that ๎‚๐‘†(๐‘Œ,๐‘)=๐‘†(๐‘Œ,๐‘)+2๐‘”(๐‘Œ,๐‘)โˆ’(๐‘›+1)๐œ‚(๐‘Œ)๐œ‚(๐‘),(3.8) where ๎‚๐‘† and ๐‘† are the Ricci tensors of the connections ๎‚โˆ‡ and โˆ‡, respectively.

Contracting (3.8), we get ฬƒ๐‘Ÿ=๐‘Ÿ+(๐‘›โˆ’1),(3.9) where ฬƒ๐‘Ÿ and ๐‘Ÿ are the scalar curvatures of the connections ๎‚โˆ‡ and โˆ‡, respectively.

4. ๐œ™-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A para-Sasakian manifold is called ๐œ™-recurrent with respect to the quarter-symmetric metric connection if its curvature tensor ๎‚๐‘… satisfies the condition ๐œ™2๎‚โˆ‡๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎‚๎‚(๐‘‹,๐‘Œ)๐‘=๐ด(๐‘Š)๐‘…(๐‘‹,๐‘Œ)๐‘.(4.1) By virtue of (2.1) and (4.1), we have ๎‚€๎‚โˆ‡๐‘Š๎‚๐‘…๎‚๎‚โˆ‡(๐‘‹,๐‘Œ)๐‘โˆ’๐œ‚๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎‚๎‚(๐‘‹,๐‘Œ)๐‘๐œ‰=๐ด(๐‘Š)๐‘…(๐‘‹,๐‘Œ)๐‘.(4.2) From which, it follows that ๐‘”๎‚โˆ‡๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎‚๎‚โˆ‡(๐‘‹,๐‘Œ)๐‘,๐‘ˆโˆ’๐œ‚๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎‚๎‚€๎‚๎‚.(๐‘‹,๐‘Œ)๐‘๐‘”(๐œ‰,๐‘ˆ)=๐ด(๐‘Š)๐‘”๐‘…(๐‘‹,๐‘Œ)๐‘,๐‘ˆ(4.3) Let {๐‘’๐‘–},๐‘–=1,2,โ€ฆ,๐‘› be an orthonormal basis of the tangent space at any point of the manifold. Then putting ๐‘‹=๐‘ˆ=๐‘’๐‘– in (4.3) and taking summation over ๐‘–,โ€‰โ€‰1โ‰ค๐‘–โ‰ค๐‘›, we get ๎‚€๎‚โˆ‡๐‘Š๎‚๐‘†๎‚(๐‘Œ,๐‘)โˆ’๐‘›๎“๐‘–=1๐œ‚๎‚โˆ‡๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎€ท๐‘’๐‘–๎€ธ๐‘๎‚๐œ‚๎€ท๐‘’,๐‘Œ๐‘–๎€ธ๎‚=๐ด(๐‘Š)๐‘†(๐‘Œ,๐‘).(4.4) The second term of (4.4) by putting ๐‘=๐œ‰ takes the form ๐‘”๎‚โˆ‡๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎€ท๐‘’๐‘–๎€ธ๎‚๎‚€๎‚โˆ‡,๐‘Œ๐œ‰,๐œ‰=๐‘”๐‘Š๎‚๐‘…๎€ท๐‘’๐‘–๎€ธ๎‚๎‚€๎‚๐‘…๎‚€๎‚โˆ‡,๐‘Œ๐œ‰,๐œ‰โˆ’๐‘”๐‘Š๐‘’๐‘–๎‚๎‚๎‚€๎‚๐‘…๎‚€๐‘’,๐‘Œ๐œ‰,๐œ‰โˆ’๐‘”๐‘–,๎‚โˆ‡๐‘Š๐‘Œ๎‚๎‚๎‚€๎‚๐‘…๎€ท๐‘’๐œ‰,๐œ‰โˆ’๐‘”๐‘–๎€ธ๎‚โˆ‡,๐‘Œ๐‘Š๎‚.๐œ‰,๐œ‰(4.5) On simplification we obtain ๐‘”๎‚โˆ‡๎‚€๎‚€๐‘Š๎‚๐‘…๎‚๎€ท๐‘’๐‘–๎€ธ๎‚,๐‘Œ๐‘,๐œ‰=0.(4.6) Therefore (4.4) can be written in the form ๎‚€๎‚โˆ‡๐‘Š๎‚๐‘†๎‚๎‚(๐‘Œ,๐‘)=๐ด(๐‘Š)๐‘†(๐‘Œ,๐‘).(4.7) Replacing ๐‘ by ๐œ‰ in the above relation, then using (3.8) and (2.9), we have๎‚€๎‚โˆ‡๐‘Š๎‚๐‘†๎‚(๐‘Œ,๐œ‰)=โˆ’2(๐‘›โˆ’1)๐ด(๐‘Š)๐œ‚(๐‘Œ).(4.8) We know that ๎‚€๎‚โˆ‡๐‘Š๎‚๐‘†๎‚๎‚โˆ‡(๐‘Œ,๐œ‰)=๐‘Š๎‚๎‚๐‘†๎‚€๎‚โˆ‡๐‘†(๐‘Œ,๐œ‰)โˆ’๐‘Š๎‚โˆ’๎‚๐‘†๎‚€๎‚โˆ‡๐‘Œ,๐œ‰๐‘Œ,๐‘Š๐œ‰๎‚.(4.9) Using (3.8), (2.6) and (2.9) in the above relation, we get ๎‚€๎‚โˆ‡๐‘Š๎‚๐‘†๎‚(๐‘Œ,๐œ‰)=โˆ’4(๐‘›โˆ’1)๐‘”(๐‘Œ,๐œ™๐‘Š)โˆ’2๐‘†(๐‘Œ,๐œ™๐‘Š)+4๐‘”(๐‘Œ,๐œ™๐‘Š).(4.10) In view of (4.8) and (4.10), we obtain โˆ’4(๐‘›โˆ’1)๐‘”(๐‘Œ,๐œ™๐‘Š)โˆ’2๐‘†(๐‘Œ,๐œ™๐‘Š)+4๐‘”(๐‘Œ,๐œ™๐‘Š)=โˆ’2(๐‘›โˆ’1)๐ด(๐‘Š)๐œ‚(๐‘Œ).(4.11) Replacing ๐‘Œ by ๐œ™๐‘Œ in (4.11) and then using (2.3) and (2.10), we have ๐‘†(๐‘Œ,๐‘Š)=โˆ’2(๐‘›โˆ’2)๐‘”(๐‘Œ,๐‘Š)+(๐‘›โˆ’3)๐œ‚(๐‘Œ)๐œ‚(๐‘Š).(4.12) Hence, we can state the following.

Theorem 4.1. If para-Sasakian manifold is ๐œ™-recurrent with respect to quarter-symmetric metric connection then it is an ๐œ‚-Einstein manifold with respect to Riemannian connection.

5. Concircular ๐œ™-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A concircular ๐œ™-recurrent para-Sasakian manifold with respect to the quarter-symmetric metric connection is defined by ๐œ™2๎‚โˆ‡๎‚ต๎‚ต๐‘Š๎‚๐ถ๎‚ถ๎‚ถ๎‚(๐‘‹,๐‘Œ)๐‘=๐ด(๐‘Š)๐ถ(๐‘‹,๐‘Œ)๐‘,(5.1) where ๎‚๐ถ is a concircular curvature tensor with respect to the quarter-symmetric metric connection given by ๎‚๎‚๐ถ(๐‘‹,๐‘Œ)๐‘=๐‘…(๐‘‹,๐‘Œ)๐‘โˆ’ฬƒ๐‘Ÿ[].๐‘›(๐‘›โˆ’1)๐‘”(๐‘Œ,๐‘)๐‘‹โˆ’๐‘”(๐‘‹,๐‘)๐‘Œ(5.2) By virtue of (2.1) and (5.1), we have ๎‚ต๎‚โˆ‡๐‘Š๎‚๐ถ๎‚ถ๎‚โˆ‡(๐‘‹,๐‘Œ)๐‘โˆ’๐œ‚๎‚ต๎‚ต๐‘Š๎‚๐ถ๎‚ถ๎‚ถ๎‚(๐‘‹,๐‘Œ)๐‘๐œ‰=๐ด(๐‘Š)๐ถ(๐‘‹,๐‘Œ)๐‘,(5.3) from which it follows that ๐‘”๎‚โˆ‡๎‚ต๎‚ต๐‘Š๎‚๐ถ๎‚ถ๎‚ถ๎‚โˆ‡(๐‘‹,๐‘Œ)๐‘,๐‘ˆโˆ’๐œ‚๎‚ต๎‚ต๐‘Š๎‚๐ถ๎‚ถ๎‚ถ๎‚ต๎‚(๐‘‹,๐‘Œ)๐‘๐‘”(๐œ‰,๐‘ˆ)=๐ด(๐‘Š)๐‘”๎‚ถ,๐ถ(๐‘‹,๐‘Œ)๐‘,๐‘ˆ(5.4) where ๎‚ต๎‚โˆ‡๐‘Š๎‚๐ถ๎‚ถโˆ‡(๐‘‹,๐‘Œ)๐‘=๎€ท๎€ท๐‘Š๐‘…๎€ธ๎€ธ[]๐œ‰[๐œ‚][][][๐œ‚]๐‘Š[][][][๐œ‚]๐œ‰[][][]+[]๐œ‰โˆ’[๐‘”][๐œ‚]โˆ’โˆ‡(๐‘‹,๐‘Œ)๐‘+6๐‘”(๐œ™๐‘Œ,๐‘)๐‘”(๐‘Š,๐‘‹)โˆ’๐‘”(๐œ™๐‘‹,๐‘)๐‘”(๐‘Š,๐‘Œ)+6(๐‘Œ)๐‘”(๐‘Š,๐‘)+๐œ‚(๐‘)๐‘”(๐‘Š,๐‘Œ)๐œ™๐‘‹โˆ’6๐œ‚(๐‘‹)๐‘”(๐‘Š,๐‘)+๐œ‚(๐‘)๐‘”(๐‘Š,๐‘‹)๐œ™๐‘Œ+2๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)โˆ’๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)๐œ™๐‘Š+6(๐‘‹)๐‘”(๐œ™๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐œ™๐‘‹,๐‘)+12๐œ‚(๐‘Š)๐œ‚(๐‘)๐œ‚(๐‘‹)๐œ™๐‘Œโˆ’๐œ‚(๐‘Œ)๐œ™๐‘‹+๐œ‚(๐‘)๐‘”(๐‘Š,๐‘Œ)๐‘‹โˆ’๐‘”(๐‘Š,๐‘‹)๐‘Œ+2๐œ‚(๐‘Š)๐œ‚(๐‘)๐œ‚(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹+12๐œ‚(๐‘Š)(๐‘Œ)๐‘”(๐œ™๐‘‹,๐‘)โˆ’๐œ‚(๐‘‹)๐‘”(๐œ™๐‘Œ,๐‘)+๐œ‚(๐‘Š)๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)โˆ’๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)๐œ‰+๐œ‚(๐‘)๐‘”(๐œ™๐‘Š,๐‘‹)๐‘Œโˆ’๐‘”(๐œ™๐‘Š,๐‘Œ)๐‘‹+๐‘”(๐‘Š,๐‘)๐œ‚(๐‘Œ)๐‘‹โˆ’๐œ‚(๐‘‹)๐‘Œ๐‘”(๐‘Š,๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐‘”(๐‘Š,๐‘Œ)๐‘”(๐‘‹,๐‘)(๐œ™๐‘Š,๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐‘”(๐œ™๐‘Š,๐‘Œ)๐‘”(๐‘‹,๐‘)๐œ‰+๐‘”(๐œ™๐‘Š,๐‘)(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹๐‘Š๐‘Ÿ[].๐‘›(๐‘›โˆ’1)๐‘”(๐‘Œ,๐‘)๐‘‹โˆ’๐‘”(๐‘‹,๐‘)๐‘Œ(5.5) Let {๐‘’๐‘–},๐‘–=1,2,โ€ฆ,๐‘› be an orthonormal basis of the tangent space at any point of the manifold. Then putting ๐‘‹=๐‘ˆ=๐‘’๐‘– in (5.4) and taking summation over ๐‘–,1โ‰ค๐‘–โ‰ค๐‘›, we get ๎€ทโˆ‡๐‘Š๐‘†๎€ธโˆ‡(๐‘Œ,๐‘)=๐‘Š๐‘Ÿ๐‘›โˆ’โˆ‡๐‘”(๐‘Œ,๐‘)+(๐‘›+4)๐œ‚(๐‘)๐‘”(๐œ™๐‘Š,๐‘Œ)+(๐‘›+3)๐œ‚(๐‘Œ)๐‘”(๐œ™๐‘Š,๐‘)+(2๐‘›โˆ’3)๐œ‚(๐‘Š)๐œ‚(๐‘Œ)๐œ‚(๐‘)โˆ’(๐‘›โˆ’1)๐œ‚(๐‘Œ)๐‘”(๐‘Š,๐‘)๐‘Š๐‘Ÿ[]๎‚ป๐‘›(๐‘›โˆ’1)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐œ‚(๐‘)+๐ด(๐‘Š)๐‘†(๐‘Œ,๐‘)โˆ’๐ด(๐‘Š)(๐‘›+1)๐œ‚(๐‘Œ)๐œ‚(๐‘)+๐‘Ÿโˆ’(๐‘›+1)๐‘›๎‚ผ.๐‘”(๐‘Œ,๐‘)(5.6) Replacing ๐‘ by ๐œ‰ in (5.6) and using (2.9), we have ๎€ทโˆ‡๐‘Š๐‘†๎€ธโˆ‡(๐‘Œ,๐œ‰)=๐‘Š๐‘Ÿ๐‘›๎‚ธ๐œ‚(๐‘Œ)+(๐‘›+4)๐‘”(๐œ™๐‘Š,๐‘Œ)+(๐‘›โˆ’2)๐œ‚(๐‘Š)๐œ‚(๐‘Œ)โˆ’๐ด(๐‘Š)๐œ‚(๐‘Œ)2๐‘›+๐‘Ÿโˆ’(๐‘›+1)๐‘›๎‚น.(5.7) We know that ๎€ทโˆ‡๐‘Š๐‘†๎€ธ(๐‘Œ,๐œ‰)=โˆ‡๐‘Š๐‘†๎€ทโˆ‡(๐‘Œ,๐œ‰)โˆ’๐‘†๐‘Š๎€ธ๎€ท๐‘Œ,๐œ‰โˆ’๐‘†๐‘Œ,โˆ‡๐‘Š๐œ‰๎€ธ.(5.8) Using (2.6) and (2.9) in the above relation, it follows that ๎€ทโˆ‡๐‘Š๐‘†๎€ธ[๐‘”](๐‘Œ,๐œ‰)=โˆ’(๐‘›โˆ’1)(๐œ™๐‘Š,๐‘Œ)โˆ’๐‘†(๐‘Œ,๐œ™๐‘Š).(5.9) In view of (5.7) and (5.9), we obtain โˆ‡๐‘†(๐‘Œ,๐œ™๐‘Š)=โˆ’(๐‘›โˆ’1)๐‘”(๐œ™๐‘Š,๐‘Œ)โˆ’๐‘Š๐‘Ÿ๐‘›๎‚ธ๐œ‚(๐‘Œ)โˆ’(๐‘›+4)๐‘”(๐œ™๐‘Š,๐‘Œ)โˆ’(๐‘›โˆ’2)๐œ‚(๐‘Š)๐œ‚(๐‘Œ)+๐ด(๐‘Š)๐œ‚(๐‘Œ)2๐‘›+๐‘Ÿโˆ’(๐‘›+1)๐‘›๎‚น.(5.10) Replacing ๐‘Œ by ๐œ™๐‘Œ in (5.10) and then using (2.3) and (2.10), we obtain ๐‘†(๐‘Œ,๐‘Š)=โˆ’(2๐‘›+3)๐‘”(๐‘Š,๐‘Œ)+(๐‘›+4)๐œ‚(๐‘Š)๐œ‚(๐‘Œ).(5.11) This leads to the following theorem.

Theorem 5.1. If para-Sasakian manifold is concircular ๐œ™-recurrent with respect to quarter-symmetric metric connection then it is an ๐œ‚-Einstein manifold with respect to Riemannian connection.

Now from (5.3), we have ๎‚ต๎‚โˆ‡๐‘Š๎‚๐ถ๎‚ถ๎‚โˆ‡(๐‘‹,๐‘Œ)๐‘=๐œ‚๎‚ต๎‚ต๐‘Š๎‚๐ถ๎‚ถ๎‚ถ๎‚(๐‘‹,๐‘Œ)๐‘๐œ‰+๐ด(๐‘Š)๐ถ(๐‘‹,๐‘Œ)๐‘.(5.12) This gives โˆ‡๎€ท๎€ท๐‘Š๐‘…๎€ธ๎€ธโˆ‡(๐‘‹,๐‘Œ)๐‘=๐œ‚๎€ท๎€ท๐‘Š๐‘…๎€ธ๎€ธ[๐œ‚][][][][][][][][]๐œ‰[][][][]๐œ‰+โˆ‡(๐‘‹,๐‘Œ)๐‘๐œ‰+6(๐‘Œ)๐‘”(๐‘Š,๐‘)โˆ’๐œ‚(๐‘)๐‘”(๐‘Š,๐‘Œ)๐œ™๐‘‹+6๐œ‚(๐‘‹)๐‘”(๐‘Š,๐‘)+๐œ‚(๐‘)๐‘”(๐‘Š,๐‘‹)๐œ™๐‘Œ+2๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)๐œ™๐‘Šโˆ’6๐œ‚(๐‘‹)๐‘”(๐œ™๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐œ™๐‘‹,๐‘)๐‘Šโˆ’2๐œ‚(๐‘Š)๐œ‚(๐‘)๐œ‚(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹+12๐œ‚(๐‘Š)๐œ‚(๐‘)๐œ‚(๐‘Œ)๐œ™๐‘‹โˆ’๐œ‚(๐‘‹)๐œ™๐‘Œโˆ’๐œ‚(๐‘)๐‘”(๐‘Š,๐‘Œ)๐‘‹โˆ’๐‘”(๐‘Š,๐‘‹)๐‘Œ+๐œ‚(๐‘)๐‘”(๐œ™๐‘Š,๐‘Œ)๐‘‹โˆ’๐‘”(๐œ™๐‘Š,๐‘‹)๐‘Œ+๐œ‚(๐‘)๐œ‚(๐‘‹)๐‘”(๐‘Š,๐‘Œ)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘Š,๐‘‹)+๐œ‚(๐‘)๐œ‚(๐‘Œ)๐‘”(๐œ™๐‘Š,๐‘‹)โˆ’๐œ‚(๐‘‹)๐‘”(๐œ™๐‘Š,๐‘Œ)๐œ‰โˆ’๐‘”(๐‘Š,๐‘)๐œ‚(๐‘Œ)๐‘‹โˆ’๐œ‚(๐‘‹)๐‘Œ+๐‘”(๐œ™๐‘Š,๐‘)๐œ‚(๐‘Œ)๐‘‹โˆ’๐œ‚(๐‘‹)๐‘Œ+6๐œ‚(๐‘Š)๐œ‚(๐‘‹)๐‘”(๐œ™๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐œ™๐‘‹,๐‘)๐‘Š๐‘Ÿ[][๐‘”][][]๐œ‰โˆ’๐‘›(๐‘›โˆ’1)๐‘”(๐‘Œ,๐‘)๐‘‹โˆ’๐‘”(๐‘‹,๐‘)๐‘Œโˆ’๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)๐œ‰+๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)๐œ‰+๐ด(๐‘Š)๐‘…(๐‘‹,๐‘Œ)๐‘+3๐ด(๐‘Š)(๐œ™๐‘‹,๐‘)๐œ™๐‘Œโˆ’๐‘”(๐œ™๐‘Œ,๐‘)๐œ™๐‘‹+๐ด(๐‘Š)๐œ‚(๐‘)๐œ‚(๐‘‹)๐‘Œโˆ’๐œ‚(๐‘Œ)๐‘‹โˆ’๐ด(๐‘Š)๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)๐‘Ÿ+(๐‘›โˆ’1)[].๐‘›(๐‘›โˆ’1)๐ด(๐‘Š)๐‘”(๐‘Œ,๐‘)๐‘‹โˆ’๐‘”(๐‘‹,๐‘)๐‘Œ(5.13) Now from (5.13) and Bianchiโ€™s second identity, we have =๐ด(๐‘Š)๐œ‚(๐‘…(๐‘‹,๐‘Œ)๐‘)+๐ด(๐‘‹)๐œ‚(๐‘…(๐‘Œ,๐‘Š)๐‘)+๐ด(๐‘Œ)๐œ‚(๐‘…(๐‘Š,๐‘‹)๐‘)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ[]+๐‘›(๐‘›โˆ’1)๐ด(๐‘Š)๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ๐‘›[]+(๐‘›โˆ’1)๐ด(๐‘‹)๐œ‚(๐‘Œ)๐‘”(๐‘Š,๐‘)โˆ’๐œ‚(๐‘Š)๐‘”(๐‘Œ,๐‘)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ[].๐‘›(๐‘›โˆ’1)๐ด(๐‘Œ)๐œ‚(๐‘Š)๐‘”(๐‘‹,๐‘)โˆ’๐œ‚(๐‘‹)๐‘”(๐‘Š,๐‘)(5.14) By virtue of (2.7), we obtain from (5.14) that [][๐‘”][]=๐ด(๐‘Š)๐‘”(๐‘‹,๐‘)๐œ‚(๐‘Œ)โˆ’๐‘”(๐‘Œ,๐‘)๐œ‚(๐‘‹)+๐ด(๐‘‹)(๐‘Œ,๐‘)๐œ‚(๐‘Š)โˆ’๐‘”(๐‘Š,๐‘)๐œ‚(๐‘Œ)+๐ด(๐‘Œ)๐‘”(๐‘Š,๐‘)๐œ‚(๐‘‹)โˆ’๐‘”(๐‘‹,๐‘)๐œ‚(๐‘Š)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ[]+๐‘›(๐‘›โˆ’1)๐ด(๐‘Š)๐œ‚(๐‘‹)๐‘”(๐‘Œ,๐‘)โˆ’๐œ‚(๐‘Œ)๐‘”(๐‘‹,๐‘)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ[]+๐‘›(๐‘›โˆ’1)๐ด(๐‘‹)๐œ‚(๐‘Œ)๐‘”(๐‘Š,๐‘)โˆ’๐œ‚(๐‘Š)๐‘”(๐‘Œ,๐‘)(๐‘›+1)(๐‘›โˆ’1)+๐‘Ÿ[].๐‘›(๐‘›โˆ’1)๐ด(๐‘Œ)๐œ‚(๐‘Š)๐‘”(๐‘‹,๐‘)โˆ’๐œ‚(๐‘‹)๐‘”(๐‘Š,๐‘)(5.15) Putting ๐‘Œ=๐‘=๐‘’๐‘– in (5.15) and taking summation over ๐‘–, 1โ‰ค๐‘–โ‰ค๐‘›, we get ๐ด(๐‘Š)๐œ‚(๐‘‹)=๐ด(๐‘‹)๐œ‚(๐‘Š),(5.16) for all vector fields ๐‘‹,๐‘Š. Replacing ๐‘‹ by ๐œ‰ in (5.16), we get ๐ด(๐‘Š)=๐œ‚(๐‘Š)๐œ‚(๐œŒ),(5.17) for any vector field ๐‘Š.

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

Theorem 5.2. In a concircular ๐œ™-recurrent para-Sasakian manifold with respect to quarter-symmetric metric connection, the characteristic vector field ๐œ‰ and the vector field ๐œŒ associated to the 1-form ๐ด are in codirectional and the 1-form ๐ด 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.