A study of W7 curvature tensor in para Kenmotsu manifolds

The object of the present paper is to study certain curvature conditions in Para Kenmotsu manifolds admitting a W7 -curvature tensor.

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Introduction

A manifold 𝑀𝑛 is called an almost para-contact metric manifolds if there exists in 𝑀𝑛 a (1,1) tensor field ϕ, a vector field ξ, and a 1-form η such that (1.1) 𝜂(𝜉) = 1, 𝜙2(𝑋) = 𝑋−𝜂(𝑋)𝜉 (1.2) 𝑔(𝑋, 𝜉) = 𝜂(𝑋), 𝑔(𝜙𝑋, 𝜙𝑌) = 𝑔(𝑋, 𝑌) −𝜂(𝑋)𝜂(𝑌) (1.3) 𝜙ξ = 0, η(ϕX) = 0, 𝑟𝑎𝑛𝑘 𝜙= 𝑛−1 Where g is the Riemmannian metric. If in addition the manifold 𝑴𝒏 satisfies (1.4) (∇𝑋𝜂)(𝑌) −(∇𝑌𝜂)𝑋= 0 (1.5) (∇𝑋∇𝑌𝜂)𝑍= [−𝑔(𝑋, 𝑍) + 𝜂(𝑋)𝜂(𝑍)]𝜂(𝑌) + [−𝑔(𝑋, 𝑌) + 𝜂(𝑋)𝜂(𝑌)]𝜂(𝑍) (1.6) ∇𝑋𝜉= 𝜙2𝑋= 𝑋−𝜂(𝑋)𝜉 Then it is called a Para-Kenmotsu manifold or briefly P-Kenmostu manifold. It is well known that is a P-Kenmotsu manifold, (1.7) 𝑆(𝑋, 𝜉) = −(𝑛−1)𝜂(𝑋) (1.8) (𝐿𝜉𝑆)(𝑋, 𝑌) = 2𝑆(𝑋, 𝑌) + 2(𝑛−1)𝜂(𝑋)𝜂(𝑌) Where 𝑆 is the Ricci tensor of type (0,2) and 𝐿 denotes the Lie derivative. 2. 𝑾𝟕 Curvature Tensor in Para Kenmotsu Manifold Mishra and Pokhariyal [2] gave the definition of 𝑊7 curvature tensor as (2.1) 𝑊7 = 𝑅(𝑋, 𝑌)𝑍+ 1 ′(𝑋, 𝑌, 𝑍, 𝑇) = 𝑅′(𝑋, 𝑌, 𝑍, 𝑇) + 𝑛−1 [𝑔(𝑌, 𝑍)𝑄𝑋−𝑅𝑖𝑐(𝑌, 𝑍)𝑋] Or (2.2) 𝑊7 𝑛−1 [𝑔(𝑌, 𝑍)𝑅𝑖𝑐(𝑋, 𝑇) −𝑅𝑖𝑐(𝑌, 𝑍)𝑔(𝑋, 𝑇) Definition 2.1 A Para Kenmotsu manifold is said to be flat if the Riemannian curvature tensor vanishes identically i.e. 𝑅(𝑋, 𝑌)𝑍= 0 Definition 2.2 A Para Kenmotsu manifold is said to be 𝑊7 flat if 𝑊7 curvature tensor vanishes identically i.e. 𝑊7(𝑋, 𝑌)𝑍= 0 Theorem 2.1 A 𝑊7 flat Para Kenmotsu manifold is a flat manifold Maths 2021; 6(5): 59-61 E Nafula School of Mathematics, University of Nairobi, Kenya Stephen K Moindi School of Mathematics, University of Nairobi, Kenya Peter W Njori School of Pure and Applied Schiences, Kirinyaga University, Kenya E Nafula School of Mathematics, University of Nairobi, Kenya http://www.mathsjournal.com Proof If a P-Kenmotsu space is 𝑊7 flat then 𝑊7 = 0. Thus from equation (2.2) (2.3) 0 = 𝑅′(𝑋, 𝑌, 𝑍, 𝑇) + 1 𝑛−1 [𝑔(𝑌, 𝑍)𝑅𝑖𝑐(𝑋, 𝑇) −𝑅𝑖𝑐(𝑌, 𝑍)𝑔(𝑋, 𝑇)] 1 1 Using equation (1.7) in (2.3) we have (2.4) 𝑅′(𝑋, 𝑌, 𝑍, 𝑇) = 𝑛−1 [𝑅𝑖𝑐(𝑌, 𝑍)𝑔(𝑋, 𝑇) −𝑔(𝑌, 𝑍)𝑅𝑖𝑐(𝑋, 𝑇) = 𝑛−1 [−(𝑛−1)𝑔(𝑌, 𝑍)𝑔(𝑋, 𝑇) + 𝑔(𝑌, 𝑍)(𝑛−1)𝑔(𝑋, 𝑇) = 𝑔(𝑌, 𝑍)𝑔(𝑋, 𝑇) −𝑔(𝑌, 𝑍)𝑔(𝑋, 𝑇) But in P-Kenmotsu manifold we have (2.5) 𝑅′(𝑋, 𝑌, 𝑍, 𝑇) = 𝑔(𝑌, 𝑍)𝑔(𝑋, 𝑇) −𝑔(𝑋, 𝑍)𝑔(𝑌, 𝑇) Equations (2.4) and (2.5) can only be equal if and only if 𝑅′(𝑋, 𝑌, 𝑍, 𝑇) = 0.

Thus also 𝑅𝑖𝑐(𝑋, 𝑌) = 0. Hence the theorem. Corollary 2.1 A 𝑊7-flat P-Kenmotsu manifold is neither Einstein nor 𝜂-Einstein. 3. A 𝑾𝟕-Semi-symmetric P-Kenmotsu manifold U.C.De and N.Guha [3] gave the definition of semi-symmetry as 𝑅(𝑋, 𝑌)𝑅(𝑍, 𝑇)𝑉= 0 Definition 3.1 A P-Kenmotsu manifold is said to be 𝑊7-semi-symmetric if 𝑅(𝑋, 𝑌)𝑊7(𝑍, 𝑇)𝑉= 0 Theorem 3.1 A 𝑊7-semi-symmetric P-Kemotsu manifold is a 𝑊7-flat manifold.

Proof If a P-Kenmotsu space is 𝑊7-semi-symmetric then 𝑅(𝑋, 𝑌)𝑊7(𝑍, 𝑇)𝑉= 0 Taking the inner product with 𝜉 𝑔(𝑅(𝑋, 𝑌)𝑊7(𝑍, 𝑇)𝑉, 𝜉) = 𝑅′(𝑋, 𝑌, 𝑊7(𝑍, 𝑇)𝑉, 𝜉) = 𝑔(𝑋, 𝜉)𝑔(𝑊7(𝑍, 𝑇)𝑉, 𝑌) −𝑔(𝑌, 𝜉)𝑔(𝑊7(𝑍, 𝑇)𝑉, 𝑋) = 𝜂(𝑋)𝑊7 ′(𝑌, 𝑍, 𝑇, 𝑉) −𝜂(𝑌)𝑊7 ′(𝑋, 𝑍, 𝑇, 𝑉) = 0 But since 𝜂(𝑋) ≠0 and 𝜂(𝑌) ≠0 we must have 𝑊7 ′(𝑌, 𝑍, 𝑇, 𝑉) = 0 and 𝑊7 ′(𝑋, 𝑌, 𝑇, 𝑉) = 0. Hence the theorem. Corollary 3.1 A 𝑊7-semi-symmetric P-Kenmostu manifold is neither Einstein nor 𝜂-Einstein. 4. A 𝑾𝟕-symmetric P-Kenmotsu manifold A P-Kenmotsu manifold is said to be 𝑊7-symmetric if ∇𝑈𝑊7(𝐴, 𝐵)𝐶= 𝑊7 ′(𝑈, 𝐴, 𝐵, 𝐶) = 0 Theorem 4.1 A 𝑊7-symmetric and 𝑊7-flat P-Kenmotsu manifold is a flat manifold.

Proof From theorem (3.1), we found out that a 𝑊7-semi-symmetric manifold is a 𝑊7-flat manifold and if a P-Kenmotsu space is 𝑊7- symmetric this implies (4.1) 𝑅(𝑋, 𝑌)𝑊7(𝐴, 𝐵)𝐶−𝑊7(𝑅(𝑋, 𝑌)𝐴, 𝐵)𝐶−𝑊7(𝐴, 𝑅(𝑋, 𝑌)𝐵)𝐶−𝑊7(𝐴, 𝐵)𝑅(𝑋, 𝑌)𝐶= 0 For each of the terms in equation (4.1), we take its inner product with 𝜉 and expand to obtain the expressions (4.2) 𝑅′(𝑋, 𝑌, 𝑊7(𝐴, 𝐵)𝐶, 𝜉) = 𝑔(𝑋, 𝜉)𝑔(𝑌, 𝑊7(𝐴, 𝐵)𝐶) −𝑔(𝑌, 𝜉)𝑔(𝑋, 𝑊7(𝐴, 𝐵)𝐶) = 𝜂(𝑋)𝑊7 ′(𝑌, 𝐴, 𝐵, 𝐶) −𝜂(𝑌)𝑊7 ′(𝑋, 𝐴, 𝐵, 𝐶) (4.3) 𝑊7 ′(𝑅(𝑋, 𝑌)𝐴, 𝐵, 𝐶, 𝜉) = 𝑅′(𝑅(𝑋, 𝑌)𝐴, 𝐵, 𝐶, 𝜉) + 1 𝑛−1 [𝑅𝑖𝑐(𝑅(𝑋, 𝑌)𝐴, 𝜉)𝑔(𝐵, 𝐶) −𝑅𝑖𝑐(𝐵, 𝐶)𝑔(𝑅(𝑋, 𝑌)𝐴, 𝜉)] Using equation (1.7) in (4.3) we have (4.4) 𝑊7 ′(𝑅(𝑋, 𝑌)𝐴, 𝐵, 𝐶, 𝜉) = 𝑅′(𝑋, 𝑌, 𝐴, 𝜉)𝑔(𝐵, 𝐶) −𝜂(𝐵)𝑅′(𝑋, 𝑌, 𝐴, 𝐶) The third term of (4.1) becomes (4.5) 𝑊7 ′(𝐴, 𝑅(𝑋, 𝑌)𝐵, 𝐶, 𝜉) = 𝑅′(𝐴, 𝑅(𝑋, 𝑌)𝐵, 𝐶, 𝜉) + 1 𝑛−1 [𝑔(𝐶, 𝑅(𝑋, 𝑌)𝐵)𝑅𝑖𝑐(𝐴, 𝜉) −𝑅𝑖𝑐(𝑅(𝑋, 𝑌)𝐵, 𝐶)𝑔(𝐴, 𝜉)] = 𝜂(𝐴)𝑅′(𝑋, 𝑌, 𝐵, 𝐶) −𝑅′(𝑋, 𝑌, 𝐵, 𝜉)𝑔(𝐴, 𝐶) Similarly, http://www.mathsjournal.com (4.6) 𝑊7 ′(𝐴, 𝐵, 𝑅(𝑋, 𝑌)𝐶, 𝜉) = 𝑅′(𝐴, 𝐵, 𝑅(𝑋, 𝑌)𝐶, 𝜉) + 1 𝑛−1 [𝑔(𝐵, 𝑅(𝑋, 𝑌)𝐶)𝑅𝑖𝑐(𝐴, 𝜉) −𝑅𝑖𝑐(𝐵, 𝑅(𝑋, 𝑌)𝐶)𝑔(𝐴, 𝜉)] = 𝜂(𝐴)𝑅′(𝑋, 𝑌, 𝐶, 𝐵) −𝜂(𝐵)𝑅′(𝑋, 𝑌, 𝐶, 𝐴) Putting together equations (4.2), (4.4), (4.5), and (4.6) we have 𝜂(𝑋)𝑊7 ′(𝑌, 𝐴, 𝐵, 𝐶) −𝜂(𝑌)𝑊7 ′(𝑋, 𝐴, 𝐵, 𝐶) −𝑅′(𝑋, 𝑌, 𝐴, 𝜉)𝑔(𝐵, 𝐶) + 𝜂(𝐵)𝑅′(𝑋, 𝑌, 𝐴, 𝐶) −𝜂(𝐴)𝑅′(𝑋, 𝑌, 𝐵, 𝐶) + 𝑅′(𝑋, 𝑌, 𝐵, 𝜉)𝑔(𝐴, 𝐶) −𝜂(𝐴)𝑅′(𝑋, 𝑌, 𝐶, 𝐵) + 𝜂(𝐵)𝑅′(𝑋, 𝑌, 𝐶, 𝐴) = 0 From the requirement that 𝑊7(𝐴, 𝐵)𝐶 be flat, the terms 𝑊7 ′(𝑌, 𝐴, 𝐵, 𝐶) and 𝑊7 ′(𝑋, 𝐴, 𝐵, 𝐶) vanish. Due to the skew-symmetric property of 𝑅′(𝑋, 𝑌, 𝐵, 𝐶) in its last two variables, the coefficients of 𝜂(𝐴) cancel out. The same applies to the coefficients of 𝜂(𝐵). We then remain with the expression 𝑅′(𝑋, 𝑌, 𝐵, 𝜉)𝑔(𝐴, 𝐶) −𝑅′(𝑋, 𝑌, 𝐴, 𝜉)𝑔(𝐵, 𝐶) = 0 Since 𝑔(𝐴, 𝐶) ≠0 and 𝑔(𝑇, 𝑉) ≠0 for arbitrary vectors 𝐴, 𝐵, 𝐶, we must have 𝑅′(𝑋, 𝑌, 𝐵, 𝜉) and 𝑅′(𝑋, 𝑌, 𝐴, 𝜉) = 0.

Hence the theorem. 5. A 𝑾𝟕-Recurrent P-Kenmotsu manifold Definition 5.1 A manifold 𝑀𝑛 is said to be a recurrent manifold [1] if (∇𝑈𝑅)(𝑋, 𝑌, 𝑍) = 𝐵(𝑈)𝑅(𝑋, 𝑌, 𝑍) Where B is the associated recurrence 1-form. Definition 5.2 Similarly, it is Ricci recurrent if (∇𝑈𝑅𝑖𝑐)(𝑋, 𝑌) = 𝐵(𝑈)𝑅𝑖𝑐(𝑋, 𝑌) Definition 5.3 We shall refer to a P-Kenmotsu manifold as 𝑊7-recurrent if (5.1) (∇𝑈𝑊7 ′(𝑋, 𝑌, 𝑍, 𝑇) = 𝐵(𝑈)𝑊7 ′(𝑋, 𝑌, 𝑍, 𝑇) Theorem 5.1 If a P-Kenmotsu manifold is 𝑊7-recurrent and Ricci recurrent, then for the same recurrence 1-form, it is recurrent. Proof From (5.1), we have (∇𝑈𝑊7 ′)(𝑋, 𝑌, 𝑍, 𝑇) = 𝐵(𝑈)𝑊7 ′(𝑋, 𝑌, 𝑍, 𝑇) = (∇𝑈𝑅′)(𝑋, 𝑌, 𝑍, 𝑇) + 1 𝑛−1 [𝑔(𝑌, 𝑍)(∇𝑈𝑅𝑖𝑐)(𝑋, 𝑇) −𝑔(𝑋, 𝑇)(∇𝑈𝑅𝑖𝑐)(𝑌, 𝑍)] ′(𝑋, 𝑌, 𝑍, 𝑇) = (∇𝑈𝑅′)(𝑋, 𝑌, 𝑍, 𝑇) + 𝐵(𝑈) 𝐵(𝑈)𝑊7 𝑛−1 [𝑔(𝑌, 𝑍)𝑅𝑖𝑐(𝑋, 𝑇) −𝑔(𝑋, 𝑇)𝑅𝑖𝑐(𝑌, 𝑍)] (∇𝑈𝑅′)(𝑋, 𝑌, 𝑍, 𝑇) = 𝐵(𝑈) {𝑊7 ′(𝑋, 𝑌, 𝑍, 𝑇) − 1 𝑛−1 [𝑔(𝑌, 𝑍)𝑅𝑖𝑐(𝑋, 𝑇) −𝑔(𝑋, 𝑇)𝑅𝑖𝑐(𝑌, 𝑍)]} (∇𝑈𝑅′)(𝑋, 𝑌, 𝑍, 𝑇) = 𝐵(𝑈)𝑅′(𝑋, 𝑌, 𝑍, 𝑇) Hence the theorem.

Conclusion

The contracted part of 𝑊7-curvature tensor does not vanish in an Einstein space. Thus we can not extend the Pirani’s formalism of gravitational wave to the Einstein space with the help of 𝑊7𝑗𝑘.

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