On Concircular 𝜙-Recurrent 𝐾-Contact Manifold Admitting Semisymmetric Metric Connection

Abstract

In the present paper, we have studied 𝜙-recurrent and concircular 𝜙-recurrent 𝐾-contact manifold with respect to semisymmetric metric connection and obtained some interesting results.

1. Introduction

The idea of semisymmetric linear connection on a differentiable manifold was introduced by Friedmann and Schouten [1]. In [2], Hayden introduced idea of metric connection with torsion on a Riemannian manifold. Further, some properties of semisymmetric metric connection has been studied by Yano [3]. In [4], Golab defined and studied quarter-symmetric connection on a differentiable manifold with affine connection, which generalizes the idea of semisymmetric connection. Various properties of semisymmetric metric connection and quarter-symmetric metric connection have been studied by many geometers like Sharfuddin and Hussain [5], Amur and Pujar [6], Rastogi [7, 8], Mishra and Pandey [9], Bagewadi et al. [10–14], De et al. [15, 16], 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 [17] introduced the notion of local 𝜙-symmetry on a Sasakian manifold. Generalizing the notion of 𝜙-symmetry, De et al. [18] introduced the notion of 𝜙-recurrent Sasakian manifolds.

The paper is organized as follows. Section 2 is devoted to preliminaries. In Section 3, we study semisymmetric metric connection in a 𝐾-contact manifold. In Section 4, it is proved that a 𝜙-recurrent 𝐾-contact manifold with respect to semisymmetric metric connection is an Einstein manifold. Finally, in Section 5 it is also shown that concircular 𝜙-recurrent 𝐾-contact manifold admitting semisymmetric metric connection is an Einstein manifold, and the characteristic vector field 𝜉 and the vector field 𝜌 associated to the 1-form 𝐴 are codirectional.

2. Preliminaries

An 𝑛-dimensional differentiable manifold 𝑀 is said to have an almost contact structure (𝜙,𝜉,𝜂) if it carries a tensor field 𝜙 of type (1,1), a vector field 𝜉, and a 1-form 𝜂 on 𝑀, respectively, such that, 𝜙²=−𝐼+𝜂⊗𝜉,𝜂(𝜉)=1,𝜂∘𝜙=0,𝜙𝜉=0. (2.1)

Thus a manifold 𝑀 equipped with this structure (𝜙,𝜉,𝜂) is called an almost contact manifold and is denoted by (𝑀,𝜙,𝜉,𝜂). If 𝑔 is a Riemannian metric on an almost contact manifold 𝑀 such that, 𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),𝑔(𝑋,𝜉)=𝜂(𝑋), (2.2) where 𝑋, 𝑌 are vector fields defined on 𝑀, then, 𝑀 is said to have an almost contact metric structure (𝜙,𝜉,𝜂,𝑔), and 𝑀 with this structure is called an almost contact metric manifold and is denoted by (𝑀,𝜙,𝜉,𝜂,𝑔).

If on (𝑀,𝜙,𝜉,𝜂,𝑔) the exterior derivative of 1-form 𝜂 satisfies 𝑑𝜂(𝑋,𝑌)=𝑔𝑋,𝜙𝑌, (2.3) then (𝜙,𝜉,𝜂,𝑔) is said to be a contact metric structure, and 𝑀 equipped with a contact metric structure is called an contact metric manifold.

If moreover 𝜉 is killing vector field on 𝑀, then, 𝑀 is called a 𝐾-contact Riemannian manifold [19, 20]. A 𝐾-contact Riemannian manifold is called Sasakian [19], if the relation ∇_𝑋𝜙𝑌=𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋 (2.4) holds, where ∇ denotes the operator of covariant differentiation with respect to 𝑔.

In a 𝐾-contact manifold 𝑀, the following relations holds: ∇_𝑋𝜉=−𝜙𝑋,𝑔(𝑅(𝑋,𝑌)𝑍,𝜉)=𝑔(𝑌,𝑍)𝜂(𝑋)−𝑔(𝑋,𝑍)𝜂(𝑌),𝑆(𝑋,𝜉)=(𝑛−1)𝜂(𝑋), (2.5)(2.6)(2.7) for all vector fields 𝑋, 𝑌, and 𝑍. Here 𝑅 and 𝑆 are the Riemannian curvature tensor and the Ricci tensor of 𝑀, respectively.

Definition 2.1. A 𝐾-contact manifold 𝑀 is said to be 𝜙-recurrent if there exists a nonzero 1-form 𝐴 such that, 𝜙²∇_𝑊𝑅(𝑋,𝑌)𝑍=𝐴(𝑊)𝑅(𝑋,𝑌)𝑍, (2.8) where 𝐴 is defined by 𝐴(𝑊)=𝑔(𝑊,𝜌), and 𝜌 is a vector field associated with the 1-form 𝐴.

Definition 2.2. A 𝐾-contact manifold 𝑀 is said to be concircular 𝜙-recurrent [12] if there exists a non-zero 1-form 𝐴 such that, 𝜙²∇_𝑊𝐶(𝑋,𝑌)𝑍=𝐴(𝑊)𝐶(𝑋,𝑌)𝑍, (2.9) where 𝐶 is a concircular curvature tensor given by [21] as follows: 𝐶()()𝑟𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍−𝑛()𝑔()()𝑌,𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍 (2.10) where 𝑅 is the Riemannian curvature tensor and 𝑟 is the scalar curvature.

A linear connection ∇ in an 𝑛-dimensional differentiable manifold 𝑀 is said to be a semisymmetric connection if its torsion tensor 𝑇 is of the form 𝑇(𝑋,𝑌)=∇_𝑋𝑌−∇_𝑌𝑋−[𝑋,𝑌]=𝜂(𝑌)𝑋−𝜂(𝑋)𝑌, (2.11) for all 𝑋, 𝑌 on 𝑇𝑀. A semisymmetric connection ∇ is called semisymmetric metric connection, if it further satisfies ∇𝑔=0.

3. Semisymmetric Metric Connection in a 𝐾-Contact Manifold

A semisymmetric metric connection ∇ in a 𝐾-contact manifold can be defined by ∇_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑌)𝑋−𝑔(𝑋,𝑌)𝜉, (3.1) where ∇ is the Levi-Civita connection on 𝑀 [3].

A relation between the curvature tensor of 𝑀, with respect to the semisymmetric metric connection ∇ and the Levi-Civita connection, ∇ is given by 𝑅()()𝑔𝑌+𝑔()()+𝑔𝑋+𝜂(𝑋)𝑔()(𝑌)𝑔(𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍+𝜙𝑌,𝑍𝑋−𝑔𝜙𝑋,𝑍𝑌,𝑍𝜙𝑋−𝑔𝑋,𝑍𝜙𝑌𝜙𝑋,𝜙𝑍𝑌−𝑔𝜙𝑌,𝜙𝑍𝑌,𝑍−𝜂𝑋,𝑍)𝜉, (3.2) where 𝑅 and 𝑅 are the Riemannian curvatures of the connections ∇ and ∇, respectively.

From (3.2), it follows that 𝑆(𝑌,𝑍)=𝑆(𝑌,𝑍)−(𝑛−2)𝑔(𝑌,𝑍)+(𝑛−2)𝑔𝜙𝑌,𝑍+(𝑛−2)𝜂(𝑌)𝜂(𝑍), (3.3) where 𝑆 and 𝑆 are the Ricci tensors of the connections ∇ and ∇, respectively.

Contracting (3.3), we get 𝑟̃=𝑟−(𝑛−1)(𝑛−2), (3.4) where 𝑟̃ and 𝑟 are the scalar curvatures of the connections ∇ and ∇, respectively.

4. 𝜙-Recurrent 𝐾-Contact Manifold with respect to Semisymmetric Metric Connection

A 𝐾-contact manifold is called 𝜙-recurrent with respect to the semisymmetric metric connection if its curvature tensor 𝑅 satisfies the following condition: 𝜙²∇_𝑊𝑅(𝑋,𝑌)𝑍=𝐴(𝑊)𝑅(𝑋,𝑌)𝑍. (4.1) By virtue of (2.1) and (4.1), we have −∇_𝑊𝑅(𝑋,𝑌)𝑍+𝜂∇_𝑊𝑅(𝑋,𝑌)𝑍𝜉=𝐴(𝑊)𝑅(𝑋,𝑌)𝑍, (4.2) from which, it follows that −𝑔∇_𝑊𝑅(𝑋,𝑌)𝑍,𝑈+𝜂∇_𝑊𝑅(𝑋,𝑌)𝑍𝑔(𝜉,𝑈)=𝐴(𝑊)𝑔𝑅(𝑋,𝑌)𝑍,𝑈. (4.3) Let {𝑒_𝑖}, 𝑖=1,2,…,𝑛 be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=𝑈=𝑒_𝑖 in (4.3) and taking summation over 𝑖, 1≤𝑖≤𝑛, we get −∇_𝑊𝑆(𝑌,𝑍)+^𝑛_𝑖=1𝜂∇_𝑊𝑅𝑒_𝑖,𝑌𝑍𝜂𝑒_𝑖=𝐴(𝑊)𝑆(𝑌,𝑍). (4.4) Put 𝑍=𝜉, then the second term of (4.4) takes the following form: 𝑔∇_𝑊𝑅𝑒_𝑖,𝑌𝜉,𝜉=𝑔∇_𝑊𝑅𝑒_𝑖,𝑌𝜉,𝜉−𝑔𝑅∇_𝑊𝑒_𝑖𝑅𝑒,𝑌𝜉,𝜉−𝑔_𝑖,∇_𝑊𝑌𝜉,𝜉−𝑔𝑅𝑒_𝑖,𝑌∇_𝑊𝜉,𝜉. (4.5) On simplification, we obtain 𝑔((∇_𝑊𝑅)(𝑒_𝑖,𝑌)𝜉,𝜉)=0.

Now (4.4) implies that ∇_𝑊𝑆(𝑌,𝜉)=−𝐴(𝑊)𝑆(𝑌,𝜉). (4.6) We know that ∇_𝑊𝑆(𝑌,𝜉)=∇_𝑊𝑆(𝑌,𝜉)−𝑆∇_𝑊𝑌,𝜉−𝑆𝑌,∇_𝑊𝜉. (4.7) Using (3.3), (2.5), and (2.7) in the above relation, we get ∇_𝑊𝑆()()−()𝑔+()𝑔()()𝑔.𝑌,𝜉=𝑆𝑌,𝜙𝑊−𝑆𝑌,𝑊𝑛−1𝑌,𝜙𝑊𝑛−1𝑌,𝑊+2𝑛−2𝜙𝑌,𝜙𝑊 (4.8) In view of (4.6) and (4.8), we have 𝑆(𝑌,𝑊)−𝑆𝑌,𝜙𝑊+(𝑛−1)𝑔𝑌,𝜙𝑊−(𝑛−1)𝑔(𝑌,𝑊)−2(𝑛−2)𝑔𝜙𝑌,𝜙𝑊=(𝑛−1)𝐴(𝑊)𝜂. (𝑌)(4.9) Again putting 𝑌=𝜙𝑌 in (4.9), we get 𝑆𝜙𝑌,𝑊−𝑆𝜙𝑌,𝜙𝑊+(𝑛−1)𝑔𝜙𝑌,𝜙𝑊−(𝑛−1)𝑔𝜙𝑌,𝑊+2(𝑛−2)𝑔𝑌,𝜙𝑊=0. (4.10) Interchanging 𝑌 and W in (4.10), we obtain 𝑆𝜙𝑊,𝑌−𝑆𝜙𝑊,𝜙𝑌+(𝑛−1)𝑔𝜙𝑊,𝜙𝑌−(𝑛−1)𝑔𝜙𝑊,𝑌+2(𝑛−2)𝑔𝑊,𝜙𝑌=0. (4.11) Adding (4.10) and (4.11) which on simplification, we have 𝑆(𝑌,𝑊)=(𝑛−1)𝑔(𝑌,𝑊). (4.12) Therefore, we can state the following.

Theorem 4.1. A 𝜙-recurrent 𝐾-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.

5. Concircular 𝜙-Recurrent 𝐾-Contact Manifold with respect to Semisymmetric Metric Connection

Let us consider a concircular 𝜙-recurrent 𝐾-contact manifold with respect to the semisymmetric metric connection defined by 𝜙²∇_𝑊𝐶(𝑋,𝑌)𝑍=𝐴(𝑊)𝐶(𝑋,𝑌)𝑍, (5.1) where 𝐶 is a concircular curvature tensor with respect to the semisymmetric metric connection given by 𝐶(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍−𝑟̃𝑛()𝑔()()𝑌.𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍 (5.2) By virtue of (2.1) and (5.1), we have −∇_𝑊𝐶(𝑋,𝑌)𝑍+𝜂∇_𝑊𝐶(𝑋,𝑌)𝑍𝜉=𝐴(𝑊)𝐶(𝑋,𝑌)𝑍, (5.3) from which, it follows that −𝑔∇_𝑊𝐶(𝑋,𝑌)𝑍,𝑈+𝜂∇_𝑊𝐶(𝑋,𝑌)𝑍𝑔(𝜉,𝑈)=𝐴(𝑊)𝑔𝐶(𝑋,𝑌)𝑍,𝑈, (5.4) where ∇_𝑊𝐶(𝑋,𝑌)𝑍=∇_𝑊𝑅()𝑍𝑔()𝜂(𝑍)()𝜂(𝑍)𝑌𝑔()𝑔()()𝑔(𝜉𝜂(𝑋)𝑔(𝑌)𝑔𝑋𝜂(𝑌)𝑔()(𝑋)𝑔(𝑔()𝜂(𝑋)()𝜂(𝑌𝑊(𝑊𝜂(𝑌)𝑔()(𝑋)𝑔(𝜉(𝑍)𝜂(𝑊𝜂(𝑋)(𝑌)𝑋(𝜂(𝑌)(𝑋)𝑌(()𝑍)(𝑋)𝑅()𝑍(𝑌)𝑅()(𝑍)𝑅()𝑊−∇𝑋,𝑌+3𝑌,𝑊𝑋−𝑔𝑋,𝑊+3𝑌,𝑍𝑊,𝑋−𝑔𝑋,𝑍𝑊,𝑌)+2𝜙𝑊,𝑍𝑌−𝜂𝜙𝑊,𝑍+2𝑋,𝑍−𝜂𝑌,𝑍)𝜙𝑊+𝑌,𝑍−𝑔𝑋,𝑍)+2𝜂)𝑋,𝑍−𝜂𝑌,𝑍)+2𝜂)𝑌−𝜂+𝑔𝑍,𝑊)𝑋−𝜂−𝑔𝑊,𝑅𝑋,𝑌𝜉−𝜂𝑊,𝑌−𝜂𝑋,𝑊𝑍−𝜂𝑋,𝑌_𝑊𝑟𝑛()𝑔()()𝑌.𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍 (5.5) Let {𝑒_𝑖}, 𝑖=1,2,…,𝑛 be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=𝑈=𝑒_𝑖 in (5.4) and taking summation over 𝑖, 1≤𝑖≤𝑛, we get ∇_𝑊𝑆()−∇𝑌,𝑍^𝑊𝑟̃𝑛𝑔()∇𝑌,𝑍=−^𝑊𝑟̃𝑛()𝑔()(𝑌)𝜂(𝑍(𝑊𝑆()−𝑛−1𝑌,𝑍−𝜂)−𝐴)𝑌,𝑍𝑟̃𝑛𝑔(𝑌,𝑍). (5.6) Replacing 𝑍 by 𝜉 in (5.6), we obtain ∇_𝑊𝑆()=∇𝑌,𝜉^𝑊𝑟̃𝑛𝜂(𝑌)−𝐴(𝑊)𝑆(𝑌,𝜉)−𝑟̃𝑛𝜂(𝑌). (5.7) We know that ∇_𝑊𝑆(𝑌,𝜉)=∇_𝑊𝑆(𝑌,𝜉)−𝑆∇_𝑊𝑌,𝜉−𝑆𝑌,∇_𝑊𝜉. (5.8) Using (3.3), (2.5) and (2.7), the above relation becomes ∇_𝑊𝑆(𝑌,𝜉)=𝑆𝑌,𝜙𝑊−𝑆(𝑌,𝑊)−(𝑛−1)𝑔𝑌,𝜙𝑊+(𝑛−1)𝑔(𝑌,𝑊)+2(𝑛−2)𝑔𝜙𝑌,𝜙𝑊. (5.9) In view of (5.7) and (5.9), we obtain 𝑆()−()𝑔+()𝑔()()𝑔()()𝜂(𝑌)𝜂=∇𝑌,𝜙𝑊−𝑆𝑌,𝑊𝑛−1𝑌,𝜙𝑊𝑛−1𝑌,𝑊+2𝑛−2𝑌,𝑊−2𝑛−2_𝑊𝑟𝑛𝜂(𝑌)(𝑊2()−𝐴)𝑛−1²−𝑟𝑛𝜂(𝑌). (𝑊)(5.10) Replacing 𝑌 by 𝜙𝑌 in (5.10), we have 𝑆𝜙𝑌,𝜙𝑊−𝑆𝜙𝑌,𝑊−(𝑛−1)𝑔𝜙𝑌,𝜙𝑊+(𝑛−1)𝑔𝜙𝑌,𝑊+2(𝑛−2)𝑔𝜙𝑌,𝑊=0. (5.11) Interchanging 𝑌 and 𝑊 in (5.11), we get 𝑆𝜙𝑊,𝜙𝑌−𝑆𝜙𝑊,𝑌−(𝑛−1)𝑔𝜙𝑊,𝜙𝑌+(𝑛−1)𝑔𝜙𝑊,𝑌+2(𝑛−2)𝑔𝜙𝑊,𝑌=0. (5.12) Adding (5.11) and (5.12), which on simplification, we have 𝑆(𝑌,𝑊)=(𝑛−1)𝑔(𝑌,𝑊). (5.13) Thus, we obtain the following theorem.

Theorem 5.1. A Concircular 𝜙-recurrent 𝐾-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.

Next, from (5.3), one has ∇_𝑊𝐶(𝑋,𝑌)𝑍=𝜂∇_𝑊𝐶(𝑋,𝑌)𝑍𝜉−𝐴(𝑊)𝐶(𝑋,𝑌)𝑍. (5.14) Now, using (3.2), (3.4), (5.5), and Bianchi's identity in (5.14), one obtains 𝐴(𝑊)𝜂(𝑅()𝑍)(𝑋)𝜂(𝑅()𝑍)(𝑌)𝜂(𝑅()𝑍)(𝑊𝑔𝜂(𝑋)𝜂(𝑌(𝑋𝑔𝜂(𝑌)𝜂(𝑊(𝑌𝑔𝜂(𝑊)𝜂(𝑋+()𝑋,𝑌+𝐴𝑌,𝑊+𝐴𝑊,𝑋=−𝐴)𝜙𝑌,𝑍−𝑔𝜙𝑋,𝑍)−𝐴)𝜙𝑊,𝑍−𝑔𝜙𝑌,𝑍)−𝐴)𝜙𝑋,𝑍−𝑔𝜙𝑊,𝑍)𝑟−𝑛−1)(𝑛−2𝑛()𝐴(𝑊𝑔()𝜂(𝑋)()𝜂(𝑌+()𝑛−1)𝑌,𝑍−𝑔𝑋,𝑍)𝑟−𝑛−1)(𝑛−2𝑛()𝐴(𝑋𝑔()𝜂(𝑌)()𝜂(𝑊+()𝑛−1)𝑊,𝑍−𝑔𝑌,𝑍)𝑟−𝑛−1)(𝑛−2𝑛()𝐴(𝑌𝑔()𝜂(𝑊)()𝜂(𝑋.𝑛−1)𝑋,𝑍−𝑔𝑊,𝑍) (5.15) Putting 𝑌=𝑍=𝑒_𝑖 in (5.15) and taking summation over 𝑖, 1≤𝑖≤𝑛, one gets −𝑛(𝑛−1)(𝑛−2)+𝑟(𝑛−2)−(𝑛−1)(𝑛−2)²𝑛()𝐴(𝑋)𝜂(𝑊)+𝑛()()+()𝑛−1𝑛−1)(𝑛−2−𝑟𝑛−2𝑛−1)(𝑛−2²𝑛()𝐴(𝑊)𝜂(𝑋)𝜂(𝑋)𝜂(𝑊).𝑛−1=𝐴𝜙𝑊−𝐴𝜙𝑋 (5.16) Replacing 𝑋 by 𝜉 in (5.16), one gets [𝑟(𝑛−2)+2(𝑛−1)(𝑛−2)]²+𝑛²(𝑛−1)²𝑛(𝑟()(𝐴(𝑊)(𝜉)𝜂(𝑊𝑛−1)[𝑛−2+2𝑛−1)(𝑛−2)]−𝐴)=0, (5.17) therefore 𝐴(𝑊)=𝜂(𝑊)𝜂𝜌, (5.18) for any vector field 𝑊.

Hence, one states the following.

Theorem 5.2. In a concircular 𝜙-recurrent 𝐾-contact manifold admitting semisymmetric metric connection the characteristic vector field 𝜉 and the vector field 𝜌 associated to the 1-form 𝐴 are co-directional and the 1-form 𝐴 is given by (5.18).

Acknowledgment

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).