## 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

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

The Riemanian curvature tensor of a generalized Sasakian space form *M*^{2n+1} (*f*_{1}, *f*_{2}, *f*_{3}) is simply given by

*f*

_{1},

*f*

_{2},

*f*

_{3}are differential functions on

*M*

^{2n+1}(

*f*

_{1},

*f*

_{2},

*f*

_{3}) and

*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].

A Riemannian manifold (*M*^{2n+1}, *g*) is called a semi-generalized recurrent manifold if its curvature tensor *R* satisfies [6, 9]

*ρ*

_{1}and

*ρ*

_{2}are two vector fields such that

*X*,

*Y*,

*Z*,

*W*and ∇ denotes the operator of covariant differentiation with respect to the metric

*g*.

A Riemannian manifold (*M*^{2n+1}, *g*) is semi generalized Ricci-recurrent if [6, 9]

*ρ*

_{1}and

*ρ*

_{2}are two vector fields such that

A Sasakian manifold (*M*^{2n+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

*X*,

*Y*,

*Z*,

*W*, where

*A*and

*B*are two non-vanishing 1-forms such that

*A*(

*X*) =

*g*(

*X*,

*ρ*

_{1}),

*B*(

*X*) =

*g*(

*X*,

*ρ*

_{2}). Here

*ρ*

_{1}and

*ρ*

_{2}are vector fields associated with 1-forms

*A*and

*B*respectively.

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

*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 (2*n* + 1) dimensional concircular curvature tensor *C* is given by [30, 31]

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

*X*and

*Y*. On a generalized Sasakian space form

*M*

^{2n+1}(

*f*

_{1},

*f*

_{2},

*f*

_{3}), we have ([1, 15])

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

## 3 Semi generalized recurrent generalized Sasakian space forms

A generalized Sasakian space form (*M*^{2n+1}, *g*) is semi-generalized recurrent manifold if

*A*and

*B*are two 1-forms,

*B*is non-zero,

*ρ*

_{1}and

*ρ*

_{2}are two vector fields such that

A generalized Sasakian space forms (*M*^{2n+1}, *g*) is semi generalized Ricci-recurrent if

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

Contracting (20) with respect to *Y*, we get

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

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

Now, we can state the following statement

*The scalar curvature r of a semi-generalized recurrent generalized Sasakian space forms is related in terms of contact forms η*(*ρ*_{1}) *and η*(*ρ*_{2}) *is given in (23).*

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

Again setting *Y* = *ξ* in (24), we get

Now, we can state the following theorem

*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

A generalized Sasakian space form (*M*^{2n+1}, *g*) is called semi-generalized *ϕ*-recurrent if its curvature tensor *R* satisfies the condition

*A*and

*B*are two 1-forms,

*B*is non-zero and these are defined by

*ρ*

_{1}and

*ρ*

_{2}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

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*in (28) and taking summation over

_{i}*i*, 1 ≤

*i*≤ (2

*n*+ 1), we get

The second term of left hand side of (29) by putting *Z* = *ξ* takes the form ((∇_{W}*R*)(*e _{i}*,

*Y*)

*Z*,

*ξ*) = 0. So, by replacing

*Z*by

*ξ*in (29) and with the help of (7) and (12), we get

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

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

*A semi generalized ϕ-recurrent generalized Sasakian space forms* (*M*^{2n+1}, *g*) *is an Einstein manifold and moreover; the 1-forms A and B are related as* −2*n*(*f*_{1} − *f*_{3})*A*(*W*) = (2*n* + 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

A generalized Sasakian space forms (*M*^{2n+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

*X*,

*Y*,

*Z*,

*W*, where

*A*and

*B*are two non-vanishing 1-forms such that

*A*(

*X*) =

*g*(

*X*,

*ρ*

_{1}),

*B*(

*X*) =

*g*(

*X*,

*ρ*

_{2}). Here

*ρ*

_{1}and

*ρ*

_{2}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

From which it follows that

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 (34) and taking summation over

*i*, 1 ≤

*i*≤ (2

*n*+ 1), and the relation

*g*((∇

_{W}

*R*)(

*X*,

*Y*)

*Z*,

*U*) = −

*g*((∇

_{W}

*R*)(

*X*,

*Y*)

*U*,

*Z*), we get

It follows that,

*K*= [(2

*n*− 1)

*B*(

*W*) −

*A*(

*W*)(

*f*

_{1}−

*f*

_{3})] and

*μ*= [(

*f*

_{1}−

*f*

_{3})

*A*(

*W*) +

*B*(

*W*)].

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

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

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

Thus we have the following assertion

*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* (*f*_{1} − *f*_{3})*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:

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

## 6 Concircularly locally *ϕ*-symmetric generalized Sasakian space forms

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

*X*,

*Y*,

*Z*are orthogonal to

*ξ*and an arbitrary vector field

*W*.

Differentiate covariantly with respect *W*, we have

Operate *ϕ*^{2} on both side, we have

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

If the manifold is conformally flat then *f*_{2} = 0. Therefore, (41) yields

Hence we can state the following theorem

*A generalized Sasakian space forms is concircularly locally ϕ-symmetric if and only if f*_{1}*and 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 M*^{2n+1} (*f*_{1}, *f*_{2}, *f*_{3}) *of dimension greater than three is conformally flat and ξ is Killing, then it is locally symmetric. Moreover, if M*^{2n+1} (*f*_{1}, *f*_{2}, *f*_{3}) *is locally symmetric, then f*_{1} − *f*_{3}*is 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 f*_{1}*and 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.*

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)).

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