Some Results on Generalized Sasakian Space Forms

Shanmukha B. Venkatesha 1
  • 1 Department of Mathematics, 577 451, Shimoga, INDIA
Shanmukha B. Venkatesha

Abstract

In the present frame work, we studied the semi generalized recurrent, semi generalized ϕ-recurrent, extended generalized ϕ-recurrent and concircularly locally ϕ-symmetric on generalized Sasakian space forms.

1 Introduction

The nature of a Riemannian manifold depends on the curvature tensor R of the manifold. It is well known that the sectional curvatures of a manifold determine its curvature tensor completely. A Riemannian manifold with constant sectional curvature c is known as a real space form and its curvature tensor is given by

R(X,Y)Z=c{g(Y,Z)Xg(X,Z)Y}.

A Sasakian manifold with constant ϕ-sectional curvature is a Sasakian space form and it has a specific form of its curvature tensor. Similar notion also holds for Kenmotsu and cosymplectic space forms. In order to generalize such space forms in a common frame Alegre, Blair and Carriazo [1] introduced and studied generalized Sasakian space forms. These space forms are defined as follows:

A generalized Sasakian space form is an almost contact metric manifold (M, ϕ, ξ, η, g), whose curvature tensor is given by

R(X,Y)Z=f1{g(Y,Z)Xg(X,Z)Y}+f2{g(X,ϕZ)ϕYg(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ}+f3{η(X)η(Z)Yη(Y)η(Z)X+g(X,Z)η(Y)ξg(Y,Z)η(X)ξ},

The Riemanian curvature tensor of a generalized Sasakian space form M2n+1 (f1, f2, f3) is simply given by

R=f1R1+f2R2+f3R3.
where f1, f2, f3 are differential functions on M2n+1 (f1, f2, f3) and
R1(X,Y)Z=g(Y,Z)Xg(X,Z)Y,R2(X,Y)Z=g(X,ϕZ)ϕYg(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ,R3(X,Y)Z=η(X)η(Z)Yη(Y)η(Z)X+g(X,Z)η(Y)ξg(Y,Z)η(X)ξ,
where f1=c+34 , f2=f3=c14 . Where c denotes the constant ϕ-sectional curvature. The properties of generalized Sasakian space form was studied by many geometers such as those mentioned in Refs. [2, 11, 12, 18, 21]. The concept of local symmetry of a Riemanian manifold has been studied by many authors in several ways to a different extent. The locally ϕ-symmetry of Sasakian manifold was introduced by Takahashi in Ref. [26]. De et.al., generalize the notion of ϕ-symmetry and then introduced the notion of ϕ-recurrent Sasakian manifold in Ref. [13]. Further ϕ-recurrent condition was studied on Kenmotsu manifold [10], LP-Sasakian manifold [27] and (LCS)n-manifold [22].

Definition 1

A Riemannian manifold (M2n+1, g) is called a semi-generalized recurrent manifold if its curvature tensor R satisfies [6, 9]

(XR)(Y,Z)W=A(X)R(Y,Z)W+B(X)g(Z,W)Y,
where A and B are two 1-forms, B is non-zero, ρ1 and ρ2 are two vector fields such that
g(X,ρ1)=A(X),g(X,ρ2)=B(X),
for any vector field X, Y, Z, W and ∇ denotes the operator of covariant differentiation with respect to the metric g.

Definition 2

A Riemannian manifold (M2n+1, g) is semi generalized Ricci-recurrent if [6, 9]

(XS)(Y,Z)=A(X)S(Y,Z)+(2n+1)B(X)g(Y,Z),
where A and B are two 1-forms, B is non-zero, ρ1 and ρ2 are two vector fields such that
g(X,ρ1)=A(X),g(X,ρ2)=B(X),

Definition 3

A Sasakian manifold (M2n+1, ϕ, ξ, η, g), n ≥ 1, is said to be an extended generalized ϕ-recurrent Sasakian manifold if its curvature tenor R satisfies the relation

ϕ2(WR)(X,Y)Z=A(W)ϕ2(R(X,Y)Z)+B(W)ϕ2(G(X,Y)Z)
for all vector fields X, Y, Z, W, where A and B are two non-vanishing 1-forms such that A(X) = g(X, ρ1), B(X) = g(X, ρ2). Here ρ1 and ρ2 are vector fields associated with 1-forms A and B respectively.

Definition 4

A generalized Sasakian space form is said to be locally ϕ-symmetric if

ϕ2(WR)(X,Y)Z=0
for all vector fields X, Y, Z orthogonal to ξ. This notion was introduced by T. Takahashi for Sasakian manifolds [26].

In 1940, Yano introduce the concircular curvature tensor. A (2n + 1) dimensional concircular curvature tensor C is given by [30, 31]

C(X,Y)Z=R(X,Y)Zr2n(2n+1){g(Y,Z)Xg(X,Z)Y},
where R and r are the Riemannian curvature tensor and scalar curvature tensor, respectively.

Author in Ref. [5] studies the symmetric conditons of generalized Sasakian space forms with concircular curvature tensor such as C(ξ, X) · C = 0, C(ξ, X) · R = 0, C(ξ, X) · S = 0 and C(ξ, X) · P = 0. Recently, researcher in Ref. [28] investigate some symmetric condition on generalized Sasakian space forms with W2-curvature tensor, such as pseudosymmetric, locally symmetric, locally ϕ-symmetric and ϕ-recurrent. Moreover many geometer’s studied the generalized Sasakian space forms with different conditions such as those mentioned in Refs. [11,12,13, 15, 16].

2 Generalized Sasakian space-forms

A (2n + 1)-dimensional Riemannian manifold is called an almost contact metric manifold if the following result holds [6], [7]:

ϕ2X=X+η(X)ξ,
η(ξ)=1,ϕξ=0,η(ϕX)=0,g(X,ξ)=η(X),
g(ϕX,ϕY)=g(X,Y)η(X)η(Y),
g(ϕX,Y)=g(X,ϕY),g(ϕX,X)=0
(Xη)(Y)=g(Xξ,Y)
for all vector field X and Y. On a generalized Sasakian space form M2n+1 (f1, f2, f3), we have ([1, 15])
(Xϕ)Y=(f1f3)(g(X,Y)ξη(Y)X),
Xξ=(f1f3)ϕX.

Again, we know that from Ref. [1], (2n + 1)-dimensional generalized Sasakian space forms holds the following relations:

S(X,Y)=(2nf1+3f2f3)g(X,Y)(3f2+(2n1)f3)η(X)η(Y),
R(X,Y)ξ=(f1f3){η(Y)Xη(X)Y},
R(ξ,X)Y=(f1f3){g(X,Y)ξη(Y)X},
η(R(X,Y)Z)=(f1f3){g(Y,Z)η(X)g(X,Z)η(Y)},
S(X,ξ)=2n(f1f3)η(X).

3 Semi generalized recurrent generalized Sasakian space forms

Definition 5

A generalized Sasakian space form (M2n+1, g) is semi-generalized recurrent manifold if

(XR)(Y,Z)W=A(X)R(Y,Z)W+B(X)g(Z,W)Y,
here A and B are two 1-forms, B is non-zero, ρ1 and ρ2 are two vector fields such that
A(X)=g(X,ρ1)andB(X)=g(X,ρ2)

Definition 6

A generalized Sasakian space forms (M2n+1, g) is semi generalized Ricci-recurrent if

(XS)(Y,Z)=A(X)S(Y,Z)+(2n+1)B(X)g(Y,Z).

Permutating equation (3) twice with respect to X, Y, Z, adding the three equations and using Bianchi second identity, we have

A(X)R(Y,Z)W+B(X)g(Z,W)+A(Y)R(Z,X)W+B(Y)g(X,W)Z+A(Z)R(X,Y)W+B(Z)g(Y,W)=0.

Contracting (20) with respect to Y, we get

A(X)S(Z,W)+B(X)g(Z,W)g(R(Z,X)ρ,W)B(Z)g(X,W)A(Z)S(X,W)+B(Z)g(X,W)=0.

Setting S(Y, Z) = g(QY, Z) in (21) and factoring off W, we get

A(X)QZ+(2n+1)B(X)ZR(Z,X)ρ+2B(Z)XA(Z)QX=0.

Again contracting with respect to Z and then substitute X = ξ in (22), one can get

r=1η(ρ1){(2n+1)22η(ρ2)2n(f1f3)[η(ρ2)+η(ρ1)]}.

Now, we can state the following statement

Theorem 1

The scalar curvature r of a semi-generalized recurrent generalized Sasakian space forms is related in terms of contact forms η(ρ1) and η(ρ2) is given in (23).

Next, we prove the semi generalized Ricci-recurrent generalized Sasakian space form, inserting Z = ξ in (19), we have

2n(f1f3)2g(W,ϕY)+(f1f3)S(Y,ϕW)=A(X)2n(f1f3)η(Y)+(2n+1)B(X)η(Y).

Again setting Y = ξ in (24), we get

A(X)2n(f1f3)+(2n+1)B(X)=0.

Now, we can state the following theorem

Theorem 2

A semi-generalized Ricci-recurrent generalized Sasakian space forms, the 1-form A and B holds (25)

4 Semi generalized ϕ-recurrent generalized Sasakian space forms

Definition 7

A generalized Sasakian space form (M2n+1, g) is called semi-generalized ϕ-recurrent if its curvature tensor R satisfies the condition

ϕ2(WR)(X,Y)Z=A(W)R(X,Y)Z+B(W)g(Y,Z)X
where A and B are two 1-forms, B is non-zero and these are defined by
A(W)=(W,ρ1),B(W)=(W,ρ2)
and ρ1 and ρ2 are vector fields associated with 1-forms A and B respectively.

Let us consider a semi-generalized ϕ-recurrent generalized Sasakian space forms. Then by virtue of (6) and (26), we have

(WR)(X,Y)Z+η((WR)(X,Y)Z)ξ=A(W)R(X,Y)Z+B(W)g(Y,Z)X.
it follows that
g((WR)(X,Y)Z,U)+η((WR)(X,Y)Z)η(U)=A(W)g(R(X,Y)Z,U)+B(W)g(Y,Z)g(X,U).

Let ei, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = ei in (28) and taking summation over i, 1 ≤ i ≤ (2n + 1), we get

(WS)(Y,Z)+i=12nη(WR)(ei,Y)Z)η(ei)=A(W)2n(f1f3)S(Y,Z)+B(W)(2n+1)g(Y,Z).

The second term of left hand side of (29) by putting Z = ξ takes the form ((∇WR)(ei, Y)Z, ξ) = 0. So, by replacing Z by ξ in (29) and with the help of (7) and (12), we get

2n(f1f3)2g(W,ϕY)+(f1f3)S(Y,ϕW)=A(W)2n(f1f3)η(Y)+B(W)(2n+1)g(Y,Z).

Inserting Y = ξ in (30) and using (7), we have

2n(f1f3)A(W)=(2n+1)B(W).

In view of (31) and replace Y by ϕY, (30) yields

S(Y,W)=2n(f1f3)g(Y,W).

Theorem 3

A semi generalized ϕ-recurrent generalized Sasakian space forms (M2n+1, g) is an Einstein manifold and moreover; the 1-forms A and B are related as −2n(f1f3)A(W) = (2n + 1)B(W).

5 Extended generalized ϕ-recurrent generalized Sasakian space forms

According to the definition of extended generalized ϕ-recurrent Sasakian manifolds, we will define the Extended generalized ϕ-recurrent generalized Sasakian space forms

Definition 8

A generalized Sasakian space forms (M2n+1, ϕ, ξ, η, g), n ≥ 1, is said to be an extended generalized ϕ-recurrent generalized Sasakian space forms if its curvature tenor R satisfies the relation

ϕ2(WR)(X,Y)Z=A(W)ϕ2(R(X,Y)Z)+B(W)ϕ2(G(X,Y)Z)
for all vector fields X, Y, Z, W, where A and B are two non-vanishing 1-forms such that A(X) = g(X, ρ1), B(X) = g(X, ρ2). Here ρ1 and ρ2 are vector fields associated with 1-forms A and B respectively.

Let us consider an extended generalized ϕ-recurrent generalized Sasakian space forms. Then by virtue of (6), we have

(WR)(X,Y)Z+η((WR)(X,Y)Z)ξ=A(W){R(X,Y)Z+η(R(X,Y)Z)}+B(W){G(X,Y)Z+η(G(X,Y)Z)}.

From which it follows that

g((WR)(X,Y)Z,U)+η((WR)(X,Y)Z)η(U)=A(W){g(R(X,Y)Z,U)+η(R(X,Y)Z)η(U)}+B(W){g(G(X,Y)Z,U)+η(G(X,Y)Z)η(U)}.

Let ei, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = ei in (34) and taking summation over i, 1 ≤ i ≤ (2n + 1), and the relation g((∇WR)(X, Y)Z, U) = −g((∇WR)(X, Y)U, Z), we get

(WS)(Y,Z)=A(W){S(Y,Z)+η(R(ξ,Y)Z)}+B(W){(2n1)g(Y,Z)η(Y)η(Z)}.

It follows that,

(WS)(Y,Z)=AS(Y,Z)+Kg(Y,Z)+μη(Y)(Z).
where K = [(2n − 1)B(W) − A(W)(f1f3)] and μ = [(f1f3)A(W) + B(W)].

Inserting Z = ξ(35) and using (12), (17) and (7), we get

2n(f1f3)2g(W,ϕY)+(f1f3)S(Y,ϕW)={2n(f1f3)A(W)+2nB(W)}η(Y).

Again inserting Y = ξ and using (7), (37) yields

2n(f1f3)A(W)+2nB(W)=0.

By taking the account of (38) in (37) and then replace Y by ϕY, we get

S(Y,W)=2n(f1f3)g(Y,W).

Thus we have the following assertion

Theorem 4

An extended generalized ϕ-recurrent generalized Sasakian space forms is an Einstein manifold and moreover the associated 1-forms A and B are related by (f1f3)A + B = 0.

It is known that a generalized Sasakian space form is Ricci-semisymmetric if and only if it is an Einstein manifold. In fact, by Theorem 4, we have the following:

Corollary 5

An extended generalized ϕ-recurrent generalized Sasakian space forms is Ricci-semisymmetric.

6 Concircularly locally ϕ-symmetric generalized Sasakian space forms

Definition 9

A (2n + 1) dimensional (n > 1) generalized Sasakian space form is called concircularly locally ϕ-symmetric if it satisfies [12].

ϕ2(WC)(X,Y)Z=0.
for all vector fields X, Y, Z are orthogonal to ξ and an arbitrary vector field W.

Differentiate covariantly with respect W, we have

(WC)(X,Y)Z=(WR)(X,Y)Zdr(W)2n(2n+1){g(Y,Z)Xg(X,Z)Y}.

Operate ϕ2 on both side, we have

ϕ2((WC)(X,Y)Z)=ϕ2((WR)(X,Y)Z)dr(W)2n(2n+1){g(Y,Z)ϕ2Xg(X,Z)ϕ2Y}.

In view of (6), and taking the help of relation (1) with X, Y, Z are orthogonal vector field, one can get

ϕ2((WC)(X,Y)Z)=df1(W){g(Y,Z)Xg(X,Z)Y}+df2(W){g(X,ϕZ)ϕYg(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ}+f2{g(X,ϕZ)(Wϕ)Y+g(X,(Wϕ)Z)ϕYg(Y,ϕZ)(Wϕ)Xg(Y,(Wϕ)Z)ϕX+2g(X,ϕY)(Wϕ)Z+2g(X,(Wϕ)Y)ϕZ}+dr(W)2n(2n+1){g(Y,Z)Xg(X,Z)Y}.

If the manifold is conformally flat then f2 = 0. Therefore, (41) yields

ϕ2((WC)(X,Y)Z)={df1(W)+dr(W)2n(2n+1)}{g(Y,Z)Xg(X,Z)Y}.

Hence we can state the following theorem

Theorem 6

A generalized Sasakian space forms is concircularly locally ϕ-symmetric if and only if f1and the scalar curvature are constant

Note 7.In [18], U. K. Kim studied generalized Sasakian space forms and proved that if a generalized Sasakian space forms M2n+1 (f1, f2, f3) of dimension greater than three is conformally flat and ξ is Killing, then it is locally symmetric. Moreover, if M2n+1 (f1, f2, f3) is locally symmetric, then f1f3is constant. In the above theorem it is shown that a conformally flat generalized Sasakian space form of dimension greater than 3 is locally ϕ-symmetric if and only if f1and scalar curvature is constant. Thus, we observe the difference between locally symmetric generalized Sasakian space forms and concircularly locally ϕ-symmetric generalized Sasakian space forms.

Acknowledgement

The first author is thankful to University Grants Commission, New Delhi, India for financial support in the form of National Fellowship for Higher Education (F1-17.1/2016-17/NFST-2015-17-STKAR-3079/(SA-III/Website)).

References

  • [1]

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

  • [2]

    P. Alegre and A. Carriazo, Structures on generalized Sasakian space forms, Differential Geom. Appl. 26 (2008), 656–666.

  • [3]

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

  • [4]

    P. Alegre and A. Carriazo, Generalized Sasakian space forms and conformal change of metric, Results Math., 59 (2011), 485–493.

  • [5]

    M. Atceken, On generalized Sasakian space forms satisfying certain condiions on the concircular curvature tensor, Bull. Math. Anal. Appl., 6 (1) (2014), 1–8.

  • [6]

    D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics Springer-Verlag, Berlin (1976) 509.

  • [7]

    D. E. Blair, Riemannian geometry of Contact and symplectic manifolds, Birkhäuser Boston, 2002.

  • [8]

    M. C. Chaki, On pseudo symmetric manifolds, Ann. St. Univ. Al I Cuza Iasi, 33 (1987).

  • [9]

    U. C. De and N. Guha, On generalized recurrent manifold, J. Nat. Acad. Math. 9 (1991), 85–92.

  • [10]

    U. C. De, A. Yildiz and A. F. Yaliniz, On ϕ-recurrent Kenmotsu manifolds, Turk J. Math., 33 (2009), 17–25.

  • [11]

    U. C. De and P. Majhi, ϕ-Semisymmetric generalized Sasakian space forms, Arab J. Math. Sci., 21 (2015), 170–178.

  • [12]

    U. C. De and A. Sarkar, Some results on generalized Sasakian-space-forms, Thai J. Math., 8 (1) (2010), 1–10.

  • [13]

    U. C. De, A. Shaikh and B. Sudipta, On ϕ-Recurrent Sasakian manifolds, Novi Sad J. Math., 33 (13) (2003), 43–48.

  • [14]

    R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44 (1992), 1–34.

  • [15]

    S. K. Hui and A. Sarkar, On the W 2 -curvature tensor of generalized Sasakian space forms, Math. Pannonica, 23 (2012), 1–12.

  • [16]

    S. K. Hui, S. Uddin and D. Chakraborty, Generalized Sasakian space forms whose metric is η-Ricci almost soliton, Differ. Geom. Dyn. Syst., 19 (2017), 45–55.

  • [17]

    S. K. Hui, D. G. Prakasha and V. Chavan, On generalized ϕ-recurrent generalized Sasakian-space-forms, Thai J. Math., 15(2), (2017) 323–332.

  • [18]

    U. K. Kim, Conformally flat generalised Sasakian space forms and locally symmetric generalized Sasakian space forms, Note di mathematica, 26 (2006), 55–67.

  • [19]

    G. P. Pokhariyal, Study of a new curvature tensor in a Sasakian manifold, Tensor N.S., 36(2) (1982), 222–225.

  • [20]

    D.G. Prakasha, On extended generalized ϕ-recurrent Sasakian manifolds, J. Egyptian Math. Soc., 21 (2013), 25–31.

  • [21]

    A. Sarkar and M. Sen, On ϕ-recurrent generalized Sasakian space forms, Lobachevskii J. Math., 33(3) (2012), 244–248.

  • [22]

    A. A. Shaikh, T. Basu and S. Eyasmin, On the existence of ϕ-recurrent (LCS)n-manifolds, Extracta Math., 23 (2008), 71–83.

  • [23]

    S. Pahana, T. Duttaa and A. Bhattacharyya, Ricci Soliton and η-Ricci Soliton on generalized Sasakian space forms, Filomat, 31(13) (2017), 4051–4062.

  • [24]

    Z. I. Szabo, Structure theorem on Riemannian space satisfying (R(X,Y) · R) = 0. I. the local version, J. Differential Geom., 17 (1982), 531–582.

  • [25]

    Shanmukha B. and Venkatesha, Some results on generalized Sasakian space forms with quarter symmetric metric connection, Asian Journal of Mathematics and Computer Research, 25 (3) 2018, 183–191.

  • [26]

    T. Takahashi, Sasakian ϕ-symmetric space, Tohoku Math. J., 29 (91) (1977), 91–113.

  • [27]

    Venkatesha and C. S. Bagewadi, On concircular ϕ-recurrent LP-Sasakian manifolds, Differ. Geom. Dyn. Syst., 10 (2008), 312–319.

  • [28]

    Venkatesha and Shanmukha B., W 2 -curvature tensor on generalized Sasakian space forms, Cubo. A Mathematical Journal. Univ. Frontera, Dep. Mat. Estad., 20 (1) 2018, 17–29.

  • [29]

    Venkatesha and Sumangala B., on M-projective curvature tensor of Generalised Sasakian space form, Acta Math., Univ. Comenianae, 2 (2013), 209–217.

  • [30]

    K. Yano, Concircular geometry I. concircular transformations, Proc. Imp. Acad. Tokyo., 16 (1940), 195–200.

  • [31]

    K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, 1953.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

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

  • [2]

    P. Alegre and A. Carriazo, Structures on generalized Sasakian space forms, Differential Geom. Appl. 26 (2008), 656–666.

  • [3]

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

  • [4]

    P. Alegre and A. Carriazo, Generalized Sasakian space forms and conformal change of metric, Results Math., 59 (2011), 485–493.

  • [5]

    M. Atceken, On generalized Sasakian space forms satisfying certain condiions on the concircular curvature tensor, Bull. Math. Anal. Appl., 6 (1) (2014), 1–8.

  • [6]

    D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics Springer-Verlag, Berlin (1976) 509.

  • [7]

    D. E. Blair, Riemannian geometry of Contact and symplectic manifolds, Birkhäuser Boston, 2002.

  • [8]

    M. C. Chaki, On pseudo symmetric manifolds, Ann. St. Univ. Al I Cuza Iasi, 33 (1987).

  • [9]

    U. C. De and N. Guha, On generalized recurrent manifold, J. Nat. Acad. Math. 9 (1991), 85–92.

  • [10]

    U. C. De, A. Yildiz and A. F. Yaliniz, On ϕ-recurrent Kenmotsu manifolds, Turk J. Math., 33 (2009), 17–25.

  • [11]

    U. C. De and P. Majhi, ϕ-Semisymmetric generalized Sasakian space forms, Arab J. Math. Sci., 21 (2015), 170–178.

  • [12]

    U. C. De and A. Sarkar, Some results on generalized Sasakian-space-forms, Thai J. Math., 8 (1) (2010), 1–10.

  • [13]

    U. C. De, A. Shaikh and B. Sudipta, On ϕ-Recurrent Sasakian manifolds, Novi Sad J. Math., 33 (13) (2003), 43–48.

  • [14]

    R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44 (1992), 1–34.

  • [15]

    S. K. Hui and A. Sarkar, On the W 2 -curvature tensor of generalized Sasakian space forms, Math. Pannonica, 23 (2012), 1–12.

  • [16]

    S. K. Hui, S. Uddin and D. Chakraborty, Generalized Sasakian space forms whose metric is η-Ricci almost soliton, Differ. Geom. Dyn. Syst., 19 (2017), 45–55.

  • [17]

    S. K. Hui, D. G. Prakasha and V. Chavan, On generalized ϕ-recurrent generalized Sasakian-space-forms, Thai J. Math., 15(2), (2017) 323–332.

  • [18]

    U. K. Kim, Conformally flat generalised Sasakian space forms and locally symmetric generalized Sasakian space forms, Note di mathematica, 26 (2006), 55–67.

  • [19]

    G. P. Pokhariyal, Study of a new curvature tensor in a Sasakian manifold, Tensor N.S., 36(2) (1982), 222–225.

  • [20]

    D.G. Prakasha, On extended generalized ϕ-recurrent Sasakian manifolds, J. Egyptian Math. Soc., 21 (2013), 25–31.

  • [21]

    A. Sarkar and M. Sen, On ϕ-recurrent generalized Sasakian space forms, Lobachevskii J. Math., 33(3) (2012), 244–248.

  • [22]

    A. A. Shaikh, T. Basu and S. Eyasmin, On the existence of ϕ-recurrent (LCS)n-manifolds, Extracta Math., 23 (2008), 71–83.

  • [23]

    S. Pahana, T. Duttaa and A. Bhattacharyya, Ricci Soliton and η-Ricci Soliton on generalized Sasakian space forms, Filomat, 31(13) (2017), 4051–4062.

  • [24]

    Z. I. Szabo, Structure theorem on Riemannian space satisfying (R(X,Y) · R) = 0. I. the local version, J. Differential Geom., 17 (1982), 531–582.

  • [25]

    Shanmukha B. and Venkatesha, Some results on generalized Sasakian space forms with quarter symmetric metric connection, Asian Journal of Mathematics and Computer Research, 25 (3) 2018, 183–191.

  • [26]

    T. Takahashi, Sasakian ϕ-symmetric space, Tohoku Math. J., 29 (91) (1977), 91–113.

  • [27]

    Venkatesha and C. S. Bagewadi, On concircular ϕ-recurrent LP-Sasakian manifolds, Differ. Geom. Dyn. Syst., 10 (2008), 312–319.

  • [28]

    Venkatesha and Shanmukha B., W 2 -curvature tensor on generalized Sasakian space forms, Cubo. A Mathematical Journal. Univ. Frontera, Dep. Mat. Estad., 20 (1) 2018, 17–29.

  • [29]

    Venkatesha and Sumangala B., on M-projective curvature tensor of Generalised Sasakian space form, Acta Math., Univ. Comenianae, 2 (2013), 209–217.

  • [30]

    K. Yano, Concircular geometry I. concircular transformations, Proc. Imp. Acad. Tokyo., 16 (1940), 195–200.

  • [31]

    K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, 1953.