Generalized Sasakian-space-forms Admitting

Generalized Sasakian-space-forms Admitting 𝑾 _𝟐 -curvature Tensor

Abhishek Kushwaha

and Dhruwa Narain

1&2

Department of Mathematics and Statistics,

D. D. U. Gorakhpur University, Gorakhpur (U.P.) – 273009, INDIA.

email:

[email protected] and

[email protected].

(Received on: June 8, 2019)

ABSTRACT

The aim of present paper is to study 𝜉−𝑊2 flat, 𝜙 − 𝑊2 flat, pseudo-𝑊2

flat and quasi−𝑊₂ flat generalized Sasakian-space-forms. Further we study 𝜙 − 𝑊₂ semi-symmetric generalized Sasakian-space-forms.

AMS Subject Classification No. 53C25, 53D15.

Keywords: Generalized Sasakian-space-forms, 𝜉 − 𝑊2 flat, 𝜙 − 𝑊2 flat, pseudo-𝑊2

flat, quasi−𝑊₂ flat and 𝜙 − 𝑊₂ semi-symmetric.

1. INTRODUCTION

In differential geometry of contact metric manifolds the nature of Riemannian manifolds depends on the curvature tensor 𝑅 of the manifolds. It is well known that the sectional curvatures of a manifold determine the curvature tensor 𝑅 completely. A Riemannian manifold with constant sectional curvature c is known as real-space-form and it’s curvature tensor 𝑅 satisfied the condition

𝑅(𝑋, 𝑌)𝑍 = 𝑐{𝑔(𝑌, 𝑍)𝑋 − 𝑔(𝑋, 𝑍)𝑌}. (1.1) Representation of these spaces are hyperbolic spaces (𝑐 < 0), the Euclidean spaces (𝑐 = 0) and the spheres (c > 0).

In contact geometry, a Sasakian manifold with constant 𝜙-sectional curvature is called Sasakian-space-forms and it has a specific form of curvature tensor. In (2004) P. Alegre, D.

Blair and A. Carriazo

introduced the notation of generalized Sasakian-space-forms. An

almost contact metric manifold (𝑀

, 𝜙, 𝜉, 𝜂, 𝑔) is said to be generalized Sasakian-space-

forms if there exist three differentiable functions 𝑓

, 𝑓

and 𝑓

on 𝑀

such that the curvature

tensor 𝑅 is given by

𝑅(𝑋, 𝑌)𝑍 = 𝑓

{𝑔(𝑌, 𝑍)𝑋 − 𝑔(𝑋, 𝑍)𝑌}

+𝑓

{𝑔(𝑋, 𝜙𝑍)𝜙𝑌 − 𝑔(𝑌, 𝜙𝑍)𝜙𝑋 + 2𝑔(𝑋, 𝜙𝑌)𝜙𝑍}

+𝑓

{𝜂(𝑋)𝜂(𝑍)𝑌 − 𝜂(𝑌)𝜂(𝑍)𝑋 + 𝑔(𝑋, 𝑍)𝜂(𝑌)𝜉 − 𝑔(𝑌, 𝑍)𝜂(𝑋)𝜉}, (1.2) for any vector fields 𝑋, 𝑌, 𝑍 on 𝑀

. In such case we denote the manifold as 𝑀

(𝑓

, 𝑓

, 𝑓

). In

the authors give several examples of generalized Sasakian-space-forms.

If 𝑓

=

, 𝑓

=

and 𝑓

=

, where c denotes 𝜙-sectional curvature, then a generalized Sasakian-space-forms with Sasakian structure becomes a Sasakian-space-forms.

A Riemannian manifold (𝑀

, 𝑔) is called locally symmetric if its curvature tensor 𝑅 is parallel, i.e., ∇𝑅 = 0, where ∇ is the Levi-Civita connection. The notation of semi-symmetric, a proper generalization of locally symmetric manifold, is defined by 𝑅(𝑋, 𝑌). 𝑅 = 0, where 𝑅(𝑋, 𝑌) acts on 𝑅 as derivation. A complete intrinsic classification of these manifolds was given by Z. I. Szabo

. Kushwaha and Narain

studied some curvature properties on Sasakian manifold. Also Kushwaha and Narain

studied projective and conformal curvature tensor in 𝐾-contact 𝜂-Einstein manifolds. In a Sasakian manifolds the Ricci operator 𝑄 commutes with 𝜙 i.e., 𝜙𝑄 = 𝑄𝜙. In

it has been shown that there exist 𝐾-contact manifold with 𝜙𝑄 = 𝑄𝜙, which is not Sasakian. It is to be noted that a 𝐾-contact manifold being intermediate between a contact metric manifold and a Sasakian manifold.

In 1970 Mishra and Pokhariyal

defined the 𝑊

-curvature tensor, given by

𝑊

(𝑋, 𝑌)𝑍 = 𝑅(𝑋, 𝑌)𝑍 +

{𝑔(𝑋, 𝑍)𝑄𝑌 − 𝑔(𝑌, 𝑍)𝑄𝑋}, (1.3) where 𝑅 is the curvature tensor and 𝑄 is the Ricci-operator, i.e.,

for all 𝑋, 𝑌.

The generalized Sasakian-space-forms have been studied by several authors

, D.

Narain, S. Yadav and Dwivedi

, De and Sarkar

, Alegre and Carriazo

and many other authors. Many geometers studied the 𝑊

-curvature tensor on different manifolds such as De and Sarkar

, A. A. Shaikh, S. K. Jana and S. Eyasmin

, Yilidiz and De

, A. Taleshian and A.

A. Husseinzadeh

and many others.

Motivated by above studies, the object of present paper is to study the properties of 𝑊

-curvature tensor in a generalized Sasakian-space-forms. The present paper is organized as follows. Section 2 is concerned with preliminaries. In section 3 we study 𝜉 − 𝑊

flat generalized Sasakian-space-forms. Section 4 is devoted to study of 𝜙 − 𝑊

flat generalized Sasakian-space-forms. Further section 5 deals with pseudo-𝑊

flat generalized Sasakian- space-forms and it is shown that pseudo-𝑊

flat generalized Sasakian-space-forms is a special type of 𝜂-Einstein manifolds. In section 6 we study quasi−𝑊

flat generalized Sasakian-space- forms and finally in last section, we study 𝜙 − 𝑊

semi-symmetric generalized Sasakian- space forms.

2. PRELIMINARIES

A (2𝑛 + 1)-dimensional Riemannian manifold 𝑀

(𝜙, 𝜉, 𝜂, 𝑔), where 𝜙 is a

tensor field, ξ is a contravariant vector field, 𝜂 is a 1-form and 𝑔 is a Riemannian metric, is

called an almost contact metric manifold if the following results holds

:

𝜙

𝑋 = −𝑋 + 𝜂(𝑋)𝜉, 𝜙𝜉 = 0, (2.1) 𝜂(𝜉) = 1, 𝑔(𝑋, 𝜉) = 𝜂(𝑋), 𝜂(𝜙𝑋) = 0, (2.2) 𝑔(𝜙𝑋, 𝜙𝑌) = 𝑔(𝑋, 𝑌) − 𝜂(𝑋)𝜂(𝑌), (2.3) 𝑔(𝜙𝑋, 𝑌) = −𝑔(𝑋, 𝜙𝑌), 𝑔(𝜙𝑋, 𝑋) = 0, (2.4) (∇

𝜂)(𝑌) = 𝑔(∇

𝜉, 𝑌). (2.5) For a (2𝑛 + 1)-dimensional generalized Sasakian-space-forms

, the curvature tensor 𝑅, the Ricci tensor 𝑆 and Ricci operator 𝑄 satisfy

𝑅(𝑋, 𝑌)𝜉 = (𝑓

− 𝑓

){𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌}, (2.6) 𝑅(𝜉, 𝑋)𝑌 = (𝑓

− 𝑓

){𝑔(𝑋, 𝑌)𝜉 − 𝜂(𝑌)𝑋}, (2.7) 𝑅(𝜉, 𝑋)𝜉 = (𝑓

− 𝑓

){𝜂(𝑋)𝜉 − 𝑋}, (2.8) 𝑆(𝑋, 𝑌) = (2𝑛𝑓

+ 3𝑓

− 𝑓

)𝑔(𝑋, 𝑌) − (3𝑓

+ (2𝑛 − 1)𝑓

)𝜂(𝑋)𝜂(𝑌), (2.9) 𝑆(𝑋, 𝜉) = 2𝑛(𝑓

− 𝑓

)𝜂(𝑋), (2.10) 𝑄𝑋 = (2𝑛𝑓

+ 3𝑓

− 𝑓

)𝑋 − (3𝑓

+ (2𝑛 − 1)𝑓

)𝜂(𝑋)𝜉, (2.11) 𝑟 = 2𝑛(2𝑛 + 1)𝑓

+ 6𝑛𝑓

− 4𝑛𝑓

, (2.12) 𝜂(𝑅(𝑋, 𝑌)𝑍) = (𝑓

− 𝑓

){𝑔(𝑌, 𝑍)𝜂(𝑋) − 𝑔(𝑋, 𝑍)𝜂(𝑌)}, (2.13) 𝜂(𝑅(𝑋, 𝑌)𝜉) = 0, (2.14) 𝜂(𝑅(𝜉, 𝑋)𝑌) = (𝑓

− 𝑓

){𝑔(𝑋, 𝑌) − 𝜂(𝑋)𝜂(𝑌)}, (2.15) 𝑆(𝜙𝑋, 𝜙𝑌) = 𝑆(𝑋, 𝑌) − 2𝑛(𝑓

− 𝑓

)𝜂(𝑋)𝜂(𝑌). (2.16) An almost contact manifold 𝑀

is said to be 𝜂-Einstein if it’s Ricci tensor 𝑆 is of the form 𝑆 = 𝐴𝑔 + 𝐵𝜂 ⊗ 𝜂, (2.17) where A and B are smooth function on the manifold, 𝜂 is called the associated 1-form and vector field 𝜉 defined by

𝑔(𝑋, 𝜉) = 𝜂(𝑋), (2.18) is called the generator. If 𝐵 = 0 then the manifold is Einstein and if 𝐴 = 0 then the manifold is special type of 𝜂-Einstein.

3. 𝝃 − 𝑾

FLAT GENERALIZED SASAKIAN SPACE FORMS

Definition 3.1. An almost contact metric manifold (𝑀

, 𝑔), 𝑛 > 1, is said to be 𝜉 − 𝑊

flat if it satisfies the condition 𝑊

(𝑋, 𝑌)𝜉 = 0 on M, for all 𝑋, 𝑌 ∈ 𝜒(𝑀).

Let M be a (2𝑛 + 1)-dimensional 𝜉 − 𝑊

flat generalized Sasakian-space-forms then from (1.3), we have

𝑅(𝑋, 𝑌)𝜉 +

{𝑔(𝑋, 𝜉)𝑄𝑌 − 𝑔(𝑌, 𝜉)𝑄𝑋} = 0. (3.1) Using (2.6) and (2.11) in (3.1), we have

(3𝑓

+ (2𝑛 − 1)𝑓

){𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌} = 0. (3.2) Since {𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌} ≠ 0, therefore (3.2) yields

𝑓

=

. (3.3) Thus we can state the following theorem:

Theorem 3.1. If a (2𝑛 + 1)-dimensional generalized Sasakian-space-forms is 𝜉 − 𝑊

flat

then 𝑓

=

.

Conversely, we assume that (3.3) holds.

Now in view of (1.3), we have

𝑊

(𝑋, 𝑌)𝜉 = 𝑅(𝑋, 𝑌)𝜉 +

{𝑔(𝑋, 𝜉)𝑄𝑌 − 𝑔(𝑌, 𝜉)𝑄𝑋}. (3.4) Using (2.6) and (2.11) in (3.1), we have

𝑊

(𝑋, 𝑌)𝜉 = [(𝑓

− 𝑓

) −

(2𝑛𝑓

+ 3𝑓

− 𝑓

)] {𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌}. (3.5) In view of (3.3) and (3.5), we obtain

𝑊

(𝑋, 𝑌)𝜉 = 0.

Hence we can state the following theorem:

Theorem 3.2. A (2𝑛 + 1)-dimensional generalized Sasakian-space-forms is 𝜉 − 𝑊

flat if and only if 𝑓

=

.

S. K. Hui and A. Sarkar

proved that a (2𝑛 + 1)-dimensional generalized Sasakian-space- forms is 𝑊

-flat if and only if 𝑓

=

.

Thus, in view of above statement and by virtue of theorem 3.2 we can state the following theorem:

Theorem 3.3. In a (2𝑛 + 1)-dimensional generalized Sasakian-space-forms following statements are equivalent:

(i) 𝑊

−flat (ii) 𝑓

=

(iii) 𝜉 − 𝑊

flat.

4. 𝝓 − 𝑾

FLAT GENERALIZED SASAKIAN-SPACE-FORMS

Definition 4.1. An almost contact metric manifold (𝑀

, 𝑔), 𝑛 > 1, is said to be 𝜙 − 𝑊

flat if it satisfies the condition

on

, for all

. Let M be a (2𝑛 + 1)-dimensional 𝜙 − 𝑊

flat generalized Sasakian-space-forms then from (1.3), we have

𝑔{𝑅((𝜙𝑋, 𝜙𝑌)𝜙𝑍, 𝜙𝑊)} +

{𝑔(𝜙𝑋, 𝜙𝑍)𝑆(𝜙𝑌, 𝜙𝑊) − 𝑔(𝜙𝑌, 𝜙𝑍)𝑆(𝜙𝑋, 𝜙𝑊)} = 0. (4.1) Using (1.2) and (2.9) in (4.1), we obtain

{𝑓

− (2𝑛𝑓

+ 3𝑓

− 𝑓

)

2𝑛 } {𝑔(𝜙𝑌, 𝜙𝑍)𝑔(𝜙𝑋, 𝜙𝑊) − 𝑔(𝜙𝑋, 𝜙𝑍)𝑔(𝜙𝑌, 𝜙𝑊)}

+𝑓

{𝑔(𝜙𝑋, 𝑍)𝑔(𝑌, 𝜙𝑊) − 𝑔(𝜙𝑌, 𝑍)𝑔(𝑋, 𝜙𝑊) + 2𝑔(𝜙𝑋, 𝑌)𝑔(𝑍, 𝜙𝑊)} = 0. (4.2) Contracting 𝑌 and 𝑍 in (4.2), we get

{𝑓

−

2𝑛

} {2𝑛𝑔(𝜙𝑋, 𝜙𝑊) − 𝑔(𝜙

𝑋, 𝜙

𝑊)} + 3𝑓

𝑔(𝜙𝑋, 𝜙𝑊) = 0. (4.3) Using (2.1) and (2.3) in (4.3), we obtain

[(2𝑛 − 1) {𝑓

−

2𝑛

} + 3𝑓

] 𝑔(𝜙𝑋, 𝜙𝑊) = 0. (4.4)

Since 𝑔(𝜙𝑋, 𝜙𝑊) ≠ 0, therefore (4.4) yields 𝑓

= 3𝑓

(1 − 2𝑛) .

Thus we can state the following theorem:

Theorem 4.1. If a (2𝑛 + 1)-dimensional generalized Sasakian-space-forms is 𝜙 − 𝑊

flat then 𝑓

=

.

5. PSEUDO - 𝑾

FLAT GENERALIZED SASAKIAN-SPACE-FORMS

Definition 5.1. An almost contact metric manifold (𝑀

, 𝑔), 𝑛 > 1, is said to be pseudo-𝑊

flat if it satisfies the condition 𝑔(𝑊

(𝜙𝑋, 𝑌)𝑍, 𝜙𝑊) = 0 on 𝑀

, for all 𝑋, 𝑌, 𝑍, 𝑊 ∈ 𝜒(𝑀).

Let us consider a (2𝑛 + 1)-dimensional pseudo-𝑊

flat generalized Sasakian-space- forms then from (1.3) we have

𝑔(𝑅(𝜙𝑋, 𝑌)𝑍, 𝜙𝑊) =

[𝑔(𝑌, 𝑍)𝑆(𝜙𝑋, 𝜙𝑊) − 𝑔(𝜙𝑋, 𝑍)𝑆(𝑌, 𝜙𝑊)]. (5.1) Let us take an orthonormal basis {𝑒

, 𝑒

, ……..,𝑒

, 𝜉} in

Putting

in (5.1), we get

∑^2𝑛_𝑖=1𝑔(𝑅(𝜙𝑋,𝑒𝑖, 𝑒𝑖, 𝜙𝑊)) = _2𝑛¹ {∑2𝑛 𝑔(𝑒𝑖, 𝑒𝑖

𝑖=1 )𝑆(𝜙𝑋, 𝜙𝑊) − ∑^2𝑛_𝑖=1𝑔(𝜙𝑋, 𝑒𝑖)𝑆(𝑒_𝑖, 𝜙𝑊)}

(5.2) In a (2𝑛 + 1)-dimensional almost contact metric manifold if {𝑒

, 𝑒

, ……..,𝑒

, 𝜉} is a local orthonormal basis of vector fields in 𝑀

then {𝜙𝑒

, 𝜙𝑒

, ……..,𝜙𝑒

, 𝜉} is also a local orthonormal basis. It is easy to verify that

∑

𝑔(𝑒

, 𝑒

) = ∑

𝑔(𝜙𝑒

, 𝜙𝑒

) = 2𝑛, (5.3)

∑

𝑔(𝜙𝑋, 𝑒

)𝑔(𝑒

, 𝜙𝑊) = ∑

𝑔(𝜙𝑋, 𝜙𝑒

)𝑔(𝜙𝑒

, 𝜙𝑊) = 𝑔(𝜙𝑋, 𝜙𝑊), (5.4)

∑

𝑔(𝑋, 𝑒

)𝑔(𝑒

, 𝜙𝑊) = ∑

𝑔(𝑋, 𝜙𝑒

)𝑔(𝜙𝑒

, 𝜙𝑊) = 𝑔(𝑋, 𝜙𝑊), (5.5)

∑

𝑔(𝜙𝑋, 𝑒

)𝑆(𝑒

, 𝜙𝑊) = 𝑆(𝜙𝑋, 𝜙𝑊). (5.6)

∑

𝑔(𝑋, 𝑒

)𝑆(𝑒

, 𝜙𝑊) = 𝑆(𝑋, 𝜙𝑊). (5.7) Using (1.2), (5.3), (5.4) and (5.6) in (5.2), we obtain

[(2𝑛 − 1)𝑓

+ 3𝑓

]𝑔(𝜙𝑋, 𝜙𝑊) = (1 −

) 𝑆(𝜙𝑋, 𝜙𝑊). (5.8) In view of (2.3), (2.16) and (5.8), we have

𝑆(𝑋, 𝑊) = 𝐴𝑔(𝑋, 𝑊) + 𝐵𝜂(𝑋)𝜂(𝑊), where 𝐴 = [2𝑛𝑓

+

(2𝑛−1)

] and 𝐵 = [−2𝑛𝑓

−

(2𝑛−1)

] . Thus we can state the following theorem:

Theorem 5.1. A (2𝑛 + 1)-dimensional pseudo−𝑊

flat generalized Sasakian-space-forms is a 𝜂-Einstein manifold.

6. QUASI - 𝑾

FLAT GENERALIZED SASAKIAN-SPACE-FORMS

Definition 6.1. An almost contact metric manifold (𝑀

, 𝑔), 𝑛 > 1, is said to be quasi-𝑊

flat if it satisfies the condition 𝑔(𝑊

(𝑋, 𝑌)𝑍, 𝜙𝑊) = 0 on 𝑀

, for all X,Y, 𝑍, 𝑊 ∈ 𝜒(𝑀).

Let us consider a (2𝑛 + 1)-dimensional quasi-𝑊

flat generalized Sasakian-space-forms then from (1.3), we have

𝑔(𝑅(𝑋, 𝑌)𝑍, 𝜙𝑊) =

[𝑔(𝑌, 𝑍)𝑆(𝑋, 𝜙𝑊) − 𝑔(𝑋, 𝑍)𝑆(𝑌, 𝜙𝑊)]. (6.1) Let us take an orthonormal basis {𝑒

, 𝑒

, ……..,𝑒

, 𝜉} in 𝑀

. Putting 𝑌 = 𝑍 = 𝑒

in (6.1), we get

∑

𝑔(𝑅(𝑋, 𝑒

, 𝑒

, 𝜙𝑊)) =

{∑

𝑔(𝑒

, 𝑒

)𝑆(𝑋, 𝜙𝑊) − ∑

𝑔(𝑋, 𝑒

)𝑆(𝑒

, 𝜙𝑊) }. (6.2) Using (1.2), (5.3), (5.5) and (5.7) in (6.2), we obtain

[(2𝑛 − 1)𝑓

+ 3𝑓

]𝑔(𝑋, 𝜙𝑊) = (1 −

) 𝑆(𝑋, 𝜙𝑊). (6.3) Replacing 𝑊 by 𝜙𝑊 in (6.3) and then using (2.1), we get

𝑆(𝑋, 𝑊) = 𝐴𝑔(𝑋, 𝑊) + 𝐵𝜂(𝑋)𝜂(𝑊), where 𝐴 = [2𝑛𝑓

+

(2𝑛−1)

] and 𝐵 = [−2𝑛𝑓

−

(2𝑛−1)

] . Thus we can state the following theorem:

Theorem 6.1. A (2𝑛 + 1)-dimensional quasi−𝑊

flat generalized Sasakian-space-forms is a 𝜂-Einstein manifold.

7. 𝜙- 𝑾

SEMI-SYMMETRIC FLAT GENERALIZED SASAKIAN-SPACE-FORMS Definition 7.1: An almost contact metric manifold (𝑀

, 𝑔), 𝑛 > 1, is said to be 𝜙-𝑊

semi-symmetric if it satisfies the condition 𝑊

(𝑋, 𝑌). 𝜙 = 0 on 𝑀

, for all 𝑋, 𝑌 ∈ 𝜒(𝑀).

Let us consider a (2𝑛 + 1)-dimensional 𝜙- 𝑊

semi-symmetric generalized Sasakian-space- forms then 𝑊

(𝑋, 𝑌). 𝜙 = 0 reduces in to

(𝑊

(𝑋, 𝑌). 𝜙)𝑍 = 𝑊

(𝑋, 𝑌)𝜙𝑍 − 𝜙𝑊

(𝑋, 𝑌)𝑍 = 0, (7.1) for any vector fields 𝑋, 𝑌 and 𝑍 ∈ 𝜒(𝑀).

Now by virtue of (1.3) and (7.1), we have

𝑅(𝑋, 𝑌)𝜙𝑍 − 𝜙𝑅(𝑋, 𝑌)𝑍 +

{𝑔(𝑋, 𝜙𝑍)𝑄𝑌 − 𝑔(𝑌, 𝜙𝑍)𝑄𝑋

−𝑔(𝑋, 𝑍)𝜙𝑄𝑌 + 𝑔(𝑌, 𝑍)𝜙𝑄𝑋} = 0. (7.2) Using (1.2), (2.1), (2.2) and (2.11) in (7.2), we have

𝑓

[𝑔(𝑌, 𝜙𝑍)𝑋 − 𝑔(𝑋, 𝜙𝑍)𝑌 − 𝑔(𝑌, 𝑍)𝜙𝑋 + 𝑔(𝑋, 𝑍)𝜙𝑌]

+ 𝑓

[−𝑔(𝑋, 𝑍)𝜙𝑌 + 𝜂(𝑋)𝜂(𝑍)𝜙𝑌 + 𝑔(𝑌, 𝑍)𝜙𝑋 − 𝜂(𝑌)𝜂(𝑍)𝜙𝑋 +𝑔(𝑋, 𝜙𝑍)𝑌 − 𝜂(𝑌)𝑔(𝑋, 𝜙𝑍)𝜉 − 𝑔(𝑌, 𝜙𝑍)𝑋 + 𝜂(𝑋)𝑔(𝑌, 𝜙𝑍)𝜉]

+𝑓

[𝑔(𝑋, 𝜙𝑍)𝜂(𝑌)𝜉 − 𝑔(𝑌, 𝜙𝑍)𝜂(𝑋)𝜉 − 𝜂(𝑋)𝜂(𝑍)𝜙𝑌 + 𝜂(𝑌)𝜂(𝑍)𝜙𝑋]

+

2𝑛

[𝑔(𝑋, 𝜙𝑍){(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝑌 − (3𝑓

+ (2𝑛 − 1)𝑓

)𝜂(𝑌)𝜉}

−𝑔(𝑌, 𝜙𝑍){(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝑋 − (3𝑓

+ (2𝑛 − 1)𝑓

)𝜂(𝑋)𝜉}

−𝑔(𝑋, 𝑍)(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝜙𝑌 + 𝑔(𝑌, 𝑍)(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝜙𝑋] = 0. (7.3) Taking 𝑌 = 𝜉 in (7.3) and using (2.1), (2.2), we get

(𝑓

− 𝑓

)[−𝑔(𝑋, 𝜙𝑍)𝜉 − 𝜂(𝑍)𝜙𝑋] + 1

2𝑛 [𝑔(𝑋, 𝜙𝑍){(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝜉

−(3𝑓

+ (2𝑛 − 1)𝑓

)𝜉} + 𝜂(𝑍)(2𝑛𝑓

+ 3𝑓

− 𝑓

)𝜙𝑋] = 0. (7.4) Inserting 𝑍 = 𝜉 in (7.4) and using (2.1), (2.2), we get

𝑓

= 3𝑓

(1 − 2𝑛) .

Thus we can state the following theorem:

Theorem 7.1: If a (2𝑛 + 1)-dimensional generalized Sasakian-space-forms is 𝜙 − 𝑊

semi- symmetric then 𝑓

=

.

ACKNOWLEDGEMENT

The first author Abhishek Kushwaha is thankful to UGC (University Grant Commission) for awarding SRF for his research work.

REFERENCES

1. A. Carriazo and P. Alegre, Generalized Sasakian-space-forms and conformal changes of the metric, Results Math., 59, 485-493 (2011).

2. A. Kushwaha and D. Narain, Some Curvature Properties on Sasakian Manifolds, J. of Int.

Academy of Physical Sciences, 20(4), 293-302 (2016).

3. A. Kushwaha and D. Narain, On K-contact 𝜂-Einstein Manifolds, J. of Rajasthan Academy of Physical Sciences, 17(3-4), 181-190 (2018).

4. A. A. Shaikh, S. K. Jana, S. Eyasmin, On weakly W

-symmetric Manifolds, Sarajevo Journal of Mathematics, 3(15) 73-91 (2007).

5. A. Yildiz and U. C. De, On a type of Kenmotsu manifolds, Diff. Geom. Dynamical system, 12, 289-298, (2010).

6. A. Taleshian and A. A. Hosseinzadeh, On W

-curvature tensor N(K)-quasi Einstein Manifolds, Int. Elec. Geom., 4, 32-47 (2011).

7. D. Narain, S. Yadav and P. K. Dwivedi, On Generalized Sasakian-space-forms Satisfying Certain Conditions, Int. J. Math and analysis, 3, 1-12 (2011).

8. D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture notes in Math. 509, Springer – Verlag, Berlin (1976).

9. G. P. Pokharial and R. S. Mishra, The Curvature tensor and their Relativistic Significance, Yokohoma Mathematicl Journal, 18, 105-108 (1970) .

10. P. Alegre, D. E. Blair and A. Carriazo, Generalized Sasakian-space-forms, Iserl. J. Math, 141, 157-183 (2004).

11. P. Alegre and A. Carriazo, Submanifolds of Generalized Sasakian-space-forms, Taiwanese J. Math., 13, 923-941 (2009).

12. P. Alegre and A. Carriazo, Structures on generalized Sasakian-spcae-forms, Diff. Geo. and

its App., 26(6), 656-666 (2008).

13. S. Sasaki, Lecture note on almost contact manifold, Part-I, Tohoku University, (1965).

14. S. Sasaki, Lecture note on almost contact manifold, Part-II, Tohoku University, (1967).

15. S. K. Hui and A. Sarkar, On the W

−Curvature Tensor of generalized Sasakian-space- forms, Mathematica Pannonica, 23(1), 113-124 (2012).

16. T. Koufogiorgos, Contact Metric Manifolds, Ann. Global Anal. Geom, 11(1), 25-34 (1993).

17. U. C. De and A. Sarkar, On the Projective Curvature Tensor of Generalized Sasakian- space-forms, Quastiones Mathematicae, 33, 245-252 (2010).

18. U. C. De and A. Sarkar, Some Results on Generalized Sasakian-space-forms, Thai J.

Math., 8, 1-10 (2010).

19. U. C. De and A. Sarkar, On a type of P-Sasakian manifolds, Maths. Reports, 11(61), 139- 144 (2009).

20. Z. I. Szabo, Structure theorem on Riemannian space Satisfying R(X, Y). R = 0, I, the local

version. J. Diff. Geom., 17, 531-582 (1982).