IOSR Journal of Engineering (IOSR-JEN) Volume 4 Issue 9 Version 4

1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 04, Issue 09 (September. 2014), ||V4|| PP 32-37
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A Quarter-Symmetric Non-Metric Connection In A Lorentzian 휷−Kenmotsu Manifold Shyam Kishor1, Prerna Kanaujia2, Satya Prakash Yadav3 1&2Department of Mathematics & Astronomy University of Lucknow-226007, India , 3SIET, Allahabad Abstract:- In this paper we study quarter- symmetric non-metric connection in a Lorentzian 훽− kenmotsu manifold and the first Bianchi identity for the curvature tensor is found. Ricci tensor and the scalar curvature with respect to quarter-symmetric non-metric connection in Lorentzian 훽−Kenmotsu manifold are obtained. Finally some identities for torsion tensor have been explored. Keywords- Hayden connection, Levi-Civita connection, Lorentzian 훽−kenmotsu manifold, quarter- symmetric metric connection, quarter symmetric non-metric connection. MSC 2010: 53B20, 53B15, 53C15.
1. INTRODUCTION
Let 푀 be an 푛−dimensional differentiable manifold equipped with a linear connection ∇ . The torsion tensor 푇 of ∇ is given by 푇 푋,푌 =훻 푋푌−훻 푌푋−[푋,푌] 푅 푋,푌 푍=훻 푋훻 푌푍−훻 푌훻 푋푍−훻 [푋,푌}푍. The connection ∇ is symmetric if its torsion tensor 푇 vanishes, otherwise it is non-symmetric. If there is a Riemannian metric 푔 in 푀 such that 훻 ̃푔 = 0, the connection ∇ is a metric connection, otherwise it is non- metric. It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection. Hayden [24] introduced a metric connection ∇ with non-zero torsion on a Riemannian manifold. Such a connection is called a Hayden connection. On the other hand, in a Riemannian manifold given a 1−form 휔, the Weyl connection ∇ constructed with  and its associated vector 퐵 (Folland 1970, [1]) is a symmetric non-metric connection. In fact, the Riemannian metric of the manifold is recurrent with respect to the Weyl connection with the recurrence 1−form , that is,∇ g=g. Another symmetric non-metric connection is projectively related to the Levi-Civita connection (cf. Yano [19], Smaranda [25]). Friedmann and Schouten ([2], [20]) introduced the idea of a semi-symmetric linear connection in a differentiable manifold. A linear connection is said to be a semi-symmetric connection if its torsion tensor 푇 is of the form 푇 (푋,푌)=푢(푌)푋−푢(푋)푌 (1.1) where 푢 is a 1−form. A Hayden connection with the torsion tensor of the form (1.1) is a semi-symmetric metric connection. In 1970, Yano [3] considered a semi-symmetric metric connection and studied some of its properties. Some different kinds of semi-symmetric connections are studied in [4], [5], [6] and [7].In 1975, S. Golab [8] defined and studied quarter-symmetric linear connections in differentiable manifolds. A linear connection is said to be a quarter-symmetric connection if its torsion tensor 푇 is of the form 푇 푋,푌 =푢 푌 휑푋−푢 푋 휑푌 ,푋,푌  푇푀 (1.2) where 푢 is a 1−form and 휑 is a tensor of type (1,1). Note that a quarter-symmetric metric connection is a Hayden connection with the torsion tensor of the form (1.2). Studies of various types of quarter-symmetric metric connections and their properties include [9], [10], [11] and [12] among others. On the other hand, there is well known class of almost contact metric manifolds introduced by K. Kenmotsu, which is now known as Kenmotsu manifolds [10]. An almost contact metric structure on a manifold 푀 is called a trans-Sasakian structure if the product manifold 푀  푅 belongs to the class 푊4. The class 퐶6 퐶5 ([13], [26]) coincides with the class of the trans-Sasakian structures of type (,). In fact, in [13], local nature of the two subclasses, namely, 퐶5 and 퐶6 structures of trans-Sasakian structures are characterized completely. We note that trans-Sasakian structures of type (0,0), (0,) and (,0) are cosymplectic [21], −Kenmotsu [14] and −Sasakian [14] respectively. The paper is organized as follows:
Section 2, deals with some preliminary results about quarter-symmetric non-metric connection. In this section the curvature tensor of the Riemannian manifold with respect to the defined quarter-symmetric non- metric connection is also found. In the last of this section first Bianchi identity for the curvature tensor of the Riemannian manifold with respect to the given quarter-symmetric non-metric connection is found. In section 3, we study this quarter-symmetric non-metric connection in Lorentzian −Kenmotsu manifold. We have given

2. A Quarter-Symmetric Non-Metric Connection In A Lorentzian 훽 −Kenmotsu Manifold
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the covariant derivative of a 1−form and the torsion tensor. We also get the curvature tensor of the Lorentzian −Kenmotsu manifold with respect to the defined quarter-symmetric non-metric connection and find first Bianchi identity. Finally we have calculated Ricci tensor, scalar curvature and torsion tensor of the Lorentzian − Kenmotsu manifold with respect to the defined quarter-symmetric non-metric connection.
2. A Quarter-Symmetric Connection
In this section existence of quarter-symmetric non-metric connection has been discussed. Theorem-1 Let 푀 be an 푛-dimensional Riemannian manifold equipped with the Levi-Civita connection ∇ of its Riemannian metric 푔. Let  be a 1-form and 휑, a (1,1) tensor field in 푀 such that  푋 =푔 ,푋 , (2.1) 푔 휑푋,푌 = −푔 푋,휑푌 (2.2) for all 푋,푌푇푀. Then there exists a unique quarter- symmetric non-metric connection ∇ in 푀 given by 훻 푋푌= ∇푋푌− 푋 휑푌−푔 푋,푌  , (2.3) That satisfies 푇 푋,푌 =  푌 휑푋− 푋 휑푌, (2.4)
and 훻 푋푔 푌,푍 = 푌 푔 푋,푍 + 푍 푔 푋,푌 (2.5) where 푇 is the torsion tensor of 훻 . Proof: The equation (2.4) of [15] is 훻 푋푌 = ∇푋푌+푢(푌)휑1푋−푢(푋)휑2푌−푔(휑1푋,푌)푈 −푓1{푢1 푋 푌+푢1 푌 푋−푔 푋,푌 푈1}−푓₂푔(푋,푌)푈₂ Taking 휑1=0,휑2=휑,푢=푢1=,푓1=0,푓2=1,푈2=, (2.6) in above equation, we get (2.3). The equations (2.5) and (2.6) of [15] are 푇 푋,푌 = 푢 푌 휑푋−푢 푋 휑푌, 훻 푋푔 푌,푍 =2푓1푢1 푋 푔 푌,푍 +푓2 푢2 푌 푔 푋,푍 +푢2 푍 푔 푋,푌 Using (2.6) in above equations, we get respectively (2.4) and (2.5). Conversely, a connection defined by (2.3) satisfies the condition (2.4) and (2.5). Proposition 1. Let 푀 be an 푛-dimensional Riemannian manifold. For the quarter-symmetric connection defined by (2.3), the covariant derivatives of the torsion tensor 푇 and any 1-form 휋 are given respectively by (훻 푋푇 )(푌,푍) = ((훻 푋휂)푍)휑푌−(훻 푋휂)푌)휑푍 +휂 푍 훻 푋휑 푌−휂 푌 훻 푋휑 푍, (2.7) and 훻 푋휋 푌= ∇푋휋 푌+ 푋 휋 휑푌 +푔 푋,푌 휋  (2.8) for all 푋,푌,푍 푇 푀. Using (2.8) & (2.3) in 훻 푋푇 푌,푍 =훻 푋푇 푌,푍 −푇 훻 푋푌,푍 −푇 (푌,훻 푋푍) We obtain (2.7). Similarly, using (2.3) with 훻 푋휋 푌=훻 푋휋푌−휋(훻 푋푌) (2.8) can be obtained. In an 푛-dimensional Riemannian manifold 푀, for the quarter-symmetric connection defined by (2.3), let us write 푇 푋,푌,푍 =푔 푇 푋,푌 ,푍 , 푋,푌,푍휖푇푀. (2.9) Proposition 2. Let 푀 be an 푛-dimensional Riemannian manifold. Then 푇 푋,푌,푍 +푇 푌,푍,푋 +푇 푍,푋,푌 =2 푋 푔 푌,휑푍 +2 푌 푔 푍,휑푋 +2 푍 푔 푋,휑푌 (2.10) Proof: In view of (2.7) and (2.9) we have the proposition. Theorem 2. Let 푀 be an 푛-dimensional Riemannian manifold equipped with the Levi-Civita connection ∇ of its Riemannian metric 푔. Then the curvature tensor 푅 of the quarter-symmetric connection defined by (2.3) is given by 푅 푋,푌,푍 =푅 푋,푌,푍 −푇 푋,푌,푍 −2푑 푋,푌 휑푍

3. A Quarter-Symmetric Non-Metric Connection In A Lorentzian 훽 −Kenmotsu Manifold
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+ 푋 ∇푌휑 푍− 푌 훻푋휑 푍 +푔 푌,푍  푋 −∇푋+ 푋 휑 − 푔 푋,푍  푌 −∇푌+ 푌 휑 (2.11) for all 푋,푌,푍푇푀, where 푅 is the curvature of Levi-Civita connection. Proof: In view of (2.3), (2.2), (2.4) and (2.9) we get (2.11). Theorem 3. In an 푛-dimensional Riemannian manifold the first Bianchi identity for the curvature tensor of the Riemannian manifold with respect to the quarter-symmetric connection defined by (2.3) is 푅 푋,푌,푍 +푅 푌,푍,푋 +푅 푍,푋,푌 =− 푇 푋,푌,푍 +푇 푌,푍,푋 +푇 푍,푋,푌  + 푋 퐵 푌,푍 + 푌 퐵 푍,푋 + 푍 퐵 푋,푌 −2푑 푋,푌 휑푍−2푑 푌,푍 휑푋−2푑 푍,푋 휑푌 (2.12) for all 푋,푌,푍 푇푀, where 퐵 푋,푌 = ∇푋휑 푌− ∇푌휑 푋. (2.13) Proof: From (2.11), we get 푅 푋,푌,푍 +푅 푌,푍,푋 +푅 푍,푋,푌 =2 푋 푔 휑푌,푍 +2 푌 푔 휑푍,푋 +2 푍 푔 휑푋,푌  + 푋 ∇푌휑 푍− 푋 훻푍휑 푌+ 푌 훻푍휑 푋 − 푌 훻푋휑 푍+ 푍 훻푋휑 푌− 푍 훻푌휑 푋 − 훻푋 푌 휑푍+ 훻푌 푋 휑푍− 훻푌 푍 휑푋 + 훻푍 푌 휑푋− 훻푍 푋 휑푌+ 훻푋 푍 휑푌. Using (2.13) and (2.10) in the previous equation we get (2.12). Let us write the curvature tensor 푇 as a (0,4) tensor by 푅 푋,푌,푍,푊 =푔 푅 푋,푌 푍,푊 , 푋,푌,푍,푊푇푀. (2.14) Then we have the following: Theorem 4.Let 푀 be a Riemannian manifold. Then 푅 푋,푌,푍,푊 +푅 푌,푋,푍,푊 =0, (2.15) for all 푋,푌,푍,푊푇푀. Proof: Using (3.25) in (2.14), we get 푅 푋,푌,푍,푊 =푅 푋,푌,푍,푊 − 푍 푔 푇 푋,푌 ,푊 +푔 푌,푍 푔 푋,푊 −푔 푋,푍 푔 푌,푊 . (2.16) Interchanging 푋 and 푌 in the previous equation and adding the resultant equation in (3.28) and using (2.4) we get (2.15).
3. QUARTER-SYMMETRIC NON-METRIC CONNECTION IN A LORENTZIAN  −KENMOTSU MANIFOLD
A differentiable manifold 푀 of dimension 푛 is called Lorentzian Kenmotsu manifold if it admits a (1,1)−tensor 휑, a contravariant vector field , a covariant vector field  and Lorentzian metric 푔 which satisfy 휑2푋=푋+ 푋 ,   =−1, 휑=0, 휑푋 =0 (3.1) 푔 휑푋,휑푌 =푔 푋,푌 + 푋  푌 (3.2) 푔 휑푋,푌 =푔 푋,휑푌 , 푔 푋, =(푋) (3.3) for all X,YTM. Also if Lorentzian Kenmotsu manifold 푀 satisfies ∇푋휑 푌 = 푔 휑푋,푌 − 푌 휑푋 , 푋,푌푇푀 (3.4) Where  denotes the operator of covariant differentiation with respect to the Lorentzian metric 푔, then 푀 is called Lorentzian -Kenmotsu manifold. From the above equation it follows that ∇푋= 푋− 푋  (3.5) (∇푋) 푌 = 푔 푋,푌 − 푋  푌 (3.6) and consequently 푑=0 (3.7) Where

4. A Quarter-Symmetric Non-Metric Connection In A Lorentzian 훽 −Kenmotsu Manifold
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푑 푋,푌 = 12 (훻푋 푌 −(훻푌) 푋 , 푋,푌푇푀 (3.8) Furthur, on a Lorentzian -Kenmotsu manifold 푀, the following relations hold ([3], [16])  푅 푋,푌 푍 =2 푔 푋,푍  푌 −푔 푌,푍  푋 (3.9) 푅 푋,푌 =2  푋 푌− 푌 푋 (3.10) 푆 푋, =− 푛−1 2 푋 (3.11) 푄 =− 푛−1 2 3.12 푆 , = 푛−1 2 (3.13) The equation (3.9) is equivalent to 푅 ,푋 푌=2  푌 푋−푔 푋,푌  (3.14) which implies that 푅 ,푋 =2 푋− 푋  (3.15) From (3.9) and (3.14), we have  푅 푋,푌  =0 (3.16)  푅 ,푋 푌 =2  푌  푋 −푔 푋,푌 . (3.17) Theorem 5. Let 푀 be a Lorentzian −Kenmotsu manifold. Then for the quarter-symmetric connection defined by (2.3), we have 훻 푋휋 푌= ∇푋휋 푌+ 푋 휋 휑푌 +푔 푋,푌 휋  . (3.18) In particular, (훻 푋휂)푌= −1 푔 푋,푌 −휂 푋 휂 푌 (3.19) and 푑 =0 (3.20) where 푑  푋,푌 = 12 훻 푋 푌 − 훻 푌 푋 , 푋,푌푇푀 (3.21) Proof: From equation (2.3), we get (3.18). Now replacing 휋 by  in (3.18) and using (3.1) and (3.6) we get (3.19). Equation (3.20) follows immediately from (3.19). Theorem 6. Let 푀 be a Lorentzian -Kenmotsu manifold. Then 훻 푋휑 푌= +1 푔 휑푋,푌 − 푌 휑푋 (3.22) which implies 푇 푋,푌 = 1  − 훻 푋휑 푌+ 훻 푌휑 푋 + (+1)  푔 휑푋,푌 −푔 휑푌,푋  (3.23) and 훻 푋=푋− +1  푋  (3.24) For all X,Y TM. Proof: From Equations (2.3) and (3.1), we get 훻 푋휑 푌= ∇푋휑 푌+푔 푋,휑푌 , Which in view of (3.4) gives (3.22). From (3.22) and (2.4), we get (3.23). Now in view of equations (2.3), (3.5), (3.1) and (3.3), we get (3.24). Theorem 7. Let 푀 be a Lorentzian 훽-Kenmotsu manifold. Then 훻 푋푇 푌,푍 = 훽+1 [푔 푋,푍 휑푌−푔 푋,푌 휑푍+ 푍 푔 휑푋,푌 − 푌 푔 휑푋,푍 ] −훽 푋 푇 푌,푍 (3.25) Consequently, 훻 푋푇 푌,푍 +훻 푌푇 푍,푋 +훻 푍푇 푋,푌 =0 (3.26) for all X,Y,Z,W∈TM. Proof: Using equations (3.19), (3.22) and (2.4) in (2.7) we obtain (3.25). Equation (3.26) follows from (3.25) and (2.4). Theorem 8. The curvature tensor 푅 of the quarter-symmetric connection in a Lorentzian 훽-kenmotsu manifold is as follows 푅 푋,푌 푍=푅 푋,푌 푍+ 훽+1 [ 푋 푔 휑푌,푍 

5. A Quarter-Symmetric Non-Metric Connection In A Lorentzian 훽 −Kenmotsu Manifold
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− 푌 푔 휑푋,푍 +푔 푌,푍  푋 −푔 푋,푍  푌 ] +훽  푍 푇 푋,푌 −푔 푌,푍 푋+푔 푋,푍 푌 (3.27) Proof: Using (3.1), (3.3), (2.9) and (2.4) in (2.11), we get 푅 푋,푌 푍=푅 푋,푌 푍−휂 푌 ∇푋휑 푍+휂 푋 ∇푌휑 푍 + (훻푌 푋)휑푍−((훻푋)푌)휑푍 +푔 푌,푍 −∇푋+ 휂 푋  −푔 푋,푍 −∇푌+ 휂 푌  −휂 푌 푔 휑푋,푍 +−휂 푋 푔 휑푌,푍 . Now using (3.4), (3.7), (3.5) and (3.1) in the above equation we obtain (3.27). Now for the curvature tensor 푅 of the quarter-symmetric non-metric connection of the Lorentzian 훽-Kenmotsu manifold we have following theorems. Theorem 9. Let 푀 be a Lorentzian 훽-Kenmotsu manifold. Then 푅 푋,푌,푍,푊 +푅 푋,푌,푊,푍 =훽 휂 푍 푔 푇 푋,푌 ,푊 +휂 푊 푔 푇 푋,푌 ,푍 +(훽+1)[{휂 푋 푔 휑푌,푍 −휂 푌 푔 휑푋,푍 +푔(푌,푍)휂(푋)−푔(푋,푍)휂(푌)}휂(푊) +{휂 푋 푔 휑푌,푊 −휂 푌 푔 휑푋,푊 +푔(푌,푊)휂(푋)−푔(푋,푊)휂(푌)}휂(푍)] (3.28) and 푅 푋,푌,푍,푊 −푅 푍,푊,푋,푌 =훽 휂 푍 푔 푇 푋,푌 ,푊 −휂 푋 푔 푇 푍,푊 ,푌 +(훽+1)[휂 푋 휂 푊 푔 휑푌,푍 +푔 푌,푍 −휂(푌)휂(푍){푔(휑푊,푋)+푔(푊,푋)}] (3.29) for all 푋,푌,푍,푊∈푇푀. Proof: From (3.27) equation (2.14) reduces to 푅 푋,푌,푍,푊 =푅(푋,푌,푍,푊)+훽[휂 푍 푔 푇 푋,푌 ,푊 −푔(푌,푍)푔(푋,푊)+푔(푋,푍)푔(푌,푊)] +(훽+1)[휂(푊){휂 푋 푔 휑푌,푍 −휂 푌 푔 휑푋,푍 +푔(푌,푍)휂(푋)−푔(푋,푍)휂(푌)}] (3.30) Interchanging 푋 and 푌 in the above equation and adding the resultant equation in it and then using (2.4) we get (2.15). Now interchanging 푍 and 푊 in (3.30) and adding the resultant equation to (3.30) we obtain (3.28). In the last the equation (3.29) can be obtained by interchanging 푋 and 푍 & 푌 and 푊 in (3.30) and substracting the resultant equation from (3.30) and using (2.4). Theorem 10: The first Bianchi identity for the curvature tensor of the Lorentzian 훽−Kenmotsu manifold with respect to the connection defined in the equation (2.3) is as given below 푅 푋,푌 푍+푅 푌,푍 푋+푅 푍,푋 푌=0 (3.31) for all 푋,푌,푍∈푇푀. Proof: In view of equations (3.27) and (2.4), we get (3.31). Theorem 11. In an 푛-dimensional Lorentzian 훽-Kenmotsu manifold 푀, the Ricci tensor and the scalar curvature with respect to the connection defined by the equation (2.3) are given by 푆 푌,푍 =푆 푌,푍 −훽 푛−1 푔 푌,푍 + 훽+1 푔 휑푌,푍 –휂 푌 푔 휑푍,휉 −휂 푌 휂 푍 (3.32) where 푋,푌∈푇푀 and 푟̃ =푟−훽푛 푛−1 − 훽+1 (3.33) respectively. Where 푆 is the Ricci tensor and 푟 is the scalar curvature of 푀. Proof: Let {e₁, e₂,……., en} be a basis of 푀, then 푆(푌,푍)=Σ푔(푅 (푒푖,푌)푍,푒푖) Now using (3.27) and trace(φ)=0 in the above equation, we obtain (3.32) and (3.32) gives (3.33). Theorem 12. The torsion tensor 푇 satisfies the following condition 푇 푇 푋,푌 푍 +푇 푇 푌,푍 푋 +푇 푇 푍,푋 푌 =0 (3.34) for all X,Y,Z∈TM Proof: Using (2.4) and (3.1) we get (3.34).

6. A Quarter-Symmetric Non-Metric Connection In A Lorentzian 훽 −Kenmotsu Manifold
International organization of Scientific Research 37 | P a g e
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