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On a class of generalized φrecurrent Sasakian manifold
Journal of the Egyptian Mathematical Society volume 27, Article number: 19 (2019)
Abstract
The object of the present paper was to introduce the notion of hyper generalized φrecurrent Sasakian manifold and quasi generalized φrecurrent Sasakian manifold and study its various geometric properties. The existence of hyper generalized φrecurrent Sasakian manifold and quasi generalized φrecurrent Sasakian manifold is proved by giving a proper example.
Introduction
The notion of contact geometry has evolved from the mathematical formalism of classical mechanics [1]. Two important classes of contact manifolds are Kcontact manifolds and Sasakian manifolds [2]. An odd dimensional analog of Kaehler geometry is the Sasakian geometry. Sasakian manifolds were firstly studied by the famous geometer Sasaki [3] in 1960 and for long time focused on this. Sasakian manifolds have been extensively studied under several points of view in [4–8] and references therein.
The notion of local symmetry of a Riemannian manifold has been weakened by several authors in many ways to a different extent. As a mild version of local symmetry, Takahashi [9] introduced the notion of local φsymmetry on a Sasakian manifold. Generalizing the idea of φsymmetry, De et al. [10] introduced the concept of φrecurrent Sasakian manifold. The notion of generalized recurrent manifolds was initiated by Dubey [11] and in [12] Shaikh et al. introduced the notion of generalized φrecurrent Sasakian manifolds. Extending the notion of generalized φrecurrent, Shaikh and Hui [13] introduced the concept of extended generalized φrecurrent manifolds. In [14], Shashikala and Venkatesha studied generalized projective φrecurrent Sasakian manifold. The extended generalized φrecurrent property in Sasakian manifold was considered by Prakasha [15] and gave some important results.
A Riemannian manifold is called generalized recurrent if its curvature tensor R satisfies the condition
where A and B are two nonvanishing 1forms defined by A(·)=g(·,γ_{1}), B(·)=g(·,γ_{2}) and the tensor P is defined by
for all X,Y,Z∈TM and ∇ denotes the covariant differentiation with respect to the metric g. Here, γ_{1} and γ_{2} are vector fields associated with 1forms A and B respectively. Especially, if the 1 form B vanishes, then (1) turns into the notion of recurrent manifold introduced by Walker [16]. A Riemannian manifold is called generalized φrecurrent if its curvature tensor R satisfies the condition
for all X,Y,Z∈TM, where P defined as in (2). Suppose the vector fields X, Y and Z are orthogonal to ξ, then the relation (3) reduces to the notion of locally generalized φrecurrent manifolds.
A Riemannian manifold is called a generalized Riccirecurrent manifold [17] if its Ricci tensor S of type (0,2) is not identically zero and satisfies the condition
where A and B are nonvanishing 1forms defined in (1). In particular, if B=0, then (4) reduces to the notion of Riccirecurrent manifold introduced by Patterson [18].
A Riemannian manifold is called a super generalized Riccirecurrent manifold if its Ricci tensor S of type (0,2) satisfies the condition
where π,ρ, and υ are nonvanishing unique 1forms. In particular, if ρ=υ, then (5) reduces to the notion of quasigeneralized Riccirecurrent manifold introduced by Shaikh and Roy [19].
Recently, Shaikh and Patra [20] introduce a generalized class of recurrent manifolds called hyper generalized recurrent manifolds. In [19], Shaikh and Roy introduce a generalized class of recurrent manifolds called quasi generalized recurrent manifolds. The present paper deals with the study of both hyper generalized φrecurrent and quasi generalized φrecurrent property in Sasakian manifolds. The paper is organized as follows: The “Preliminaries” section is concerned with some preliminaries about Sasakian manifolds. In the “Hyper generalized φrecurrent manifold,” we introduce an extended form of hyper generalized recurrent manifolds called hyper generalized φrecurrent manifolds. We study some geometric properties of this in Sasakian manifold and obtained some interesting results. We construct a proper example of a hyper generalized φrecurrent Sasakian manifold which is neither φsymmetric nor φrecurrent in the “Example of hyper generalized φrecurrent Sasakian manifold” section. In the “Quasi generalized φrecurrent manifold” section, we introduce a generalized class of φrecurrent manifold called quasi generalized φrecurrent manifold and we study this property in Sasakian manifold and obtained some interesting results. Also, the existence of quasi generalized φrecurrent Sasakian manifold is ensured by a proper example in the last section.
Preliminaries
In this section, we provide some general definition and basic formulas on contact metric manifolds and Sasakian manifolds which we will use in further sections. We may refer to [21–23] and references therein for more details and information about Sasakian geometry.
A (2n+1)dimensional smooth connected manifold M is called almost contact manifold if it admits a triple (φ,ξ,η), where φ is tensor field of type (1,1), ξ is a global vector field and η is a 1form, such that
for all X,Y∈TM. If an almost contact manifold M admits a (φ,ξ,η,g), g being a Riemannian metric such that
then M is called an almost contact metric manifold. An almost contact metric manifold M(φ,ξ,η,g) with dη(X,Y)=Φ(X,Y), Φ being the fundamental 2form of M(φ,ξ,η,g) and is defined by Φ(X,Y)=g(X,φY), is a contact metric manifold and g is the associated metric. If, in addition ξ is a Killing vector field (equivalentely, \(h=\frac {1}{2}L_{\xi } \varphi =0\), where L denotes Lie differentiation), then the manifold is called Kcontact manifold. It is well known that [2], if the contact metric structure (φ,ξ,η,g) is normal, that is, [φ,φ]+2dη⊗ξ=0 holds, then (φ,ξ,η,g) is Sasakian. An almost contact metric manifold is Sasakian if and only if
for all vector fields X and Y on M, where ∇ is LeviCivita connection of g. A Sasakian manifold is always a Kcontact manifold. The converse also holds when the dimension is three, but which may not be true in higher dimensions [24]. On any Sasakian manifold, the following relations are well known;
for all X,Y∈TM, where R, S, and Q denotes the curvature tensor, Ricci tensor and Ricci operator respectively.
Definition 1
A (2n+1)dimensional Sasakian manifold M is said to be ηEinstein if its Ricci tensor S is of the form
for any vector fields X and Y, where a and b are constants. If b=0, then the manifold M is an Einstein manifold.
Hyper generalized φrecurrent Sasakian manifold
Recently, the authors [20] studied hyper generalized recurrent manifolds and obtained several interesting results. By observing this work, we extend the notion called hyper generalized φrecurrent manifolds. In this section, we study hyper generalized φrecurrent Sasakian manifolds.
Definition 2
A Sasakian manifold M is said to be a hyper generalized φrecurrent Sasakian manifold if its curvature tensor R satisfies the condition
for all X,Y,Z∈TM, where A and B are two nonvanishing 1forms such that A(X)=g(X,ρ_{1}), B(X)=g(X,ρ_{2}) and the tensor H is defined by
for all X,Y,Z∈TM. Here, ρ_{1} and ρ_{2} are vector fields associated with 1forms A and B respectively. Especially, if the 1form B vanishes, then (15) turns into the notion of φrecurrent manifold.
Now we prove the following;
Theorem 1
Let M be a hyper generalized φrecurrent Sasakian manifold.

If the scalar curvature is zero everywhere on M, then M is Ricci recurrent.

If the scalar curvature is nonzero everywhere on M, then M is generalized Ricci recurrent.
Proof
Let us consider hyper generalized φrecurrent Sasakian manifold. In view of (6), Eq. (15) gives
this can be written as
Let \(\{e_{i}\}_{i=1}^{2n+1}\) be an orthonormal basis of the manifold. Plugging X=U=e_{i} in (18) and taking summation over i, 1≤i≤2n+1, and then using (16), we get
The second term of left hand side in (19) reduces to
Using (9), (10) and the relation g((∇_{W}R)(X,Y)Z,U)=−g((∇_{W}R)(X,Y)U,Z), we get
By virtue of (20) and (21), it follows from (19) that
where T(W)=−(A(W)+(2n−1)B(W)) and Ψ(W)=−rB(W). In the above equation, we have hyper generalized φrecurrent Sasakian manifold is Ricci recurrent (respectively generalized Ricci recurrent) if the scalar curvature is zero (respectively nonzero) everywhere on M. This completes the proof. □
Theorem 2
A hyper generalized φrecurrent Sasakian manifold M with non vanishing scalar curvature is an Einstein manifold and moreover the associated vector fields ρ_{1} and ρ_{2} of the 1forms A and B respectively are codirectional.
Proof
Taking Z=ξ in (22) and then using first term of (13), we obtain
At this point, we note that
In view of (9) and first term of (13) in (24), it follows that
Comparing (23) and (25), we get
Again taking φY instead of Y in (26) and using (6), (7) and (14), we have
Substituting Y by ξ in (26), we get
Contracting (27) over Y and W, we get
In view of (28) and (29), we have
From (27) and (30), the theorem follows. □
It is well known that a Sasakian manifold is Riccisemisymmetric if and only if it is an Einstein manifold. In fact, by Theorem 2, we have the following;
Corollary 2.1
A hyper generalized φrecurrent Sasakian manifold with non vanishing scalar curvature is Riccisemisymmetric.
Next, in a Sasakian manifold it can be easily verify that
By virtue of (12), it follows from (31) that
It is well known that in a Sasakian manifold the following relation holds [8];
for any X,Y,Z∈TM. In view of (31) and (33), it follows that
In view of (32) and (34), we obtain from (17) that
In view of (10), (27) and (30), the above equation becomes
Operating φ on both sides of (36) and using (6), we get
Hence, we can state the following;
Theorem 3
A hyper generalized φrecurrent Sasakian manifold of non vanising scalar curvature is a space of constant curvature +1.
Example of a hyper generalized φrecurrent Sasakian manifold
In this section we give an example of a hyper generalized φrecurrent Sasakian manifold. We consider threedimensional manifold M={(x,y,z)∈R^{3},(x,y,z)≠(0,0,0)}, where (x,y,z) are the standard coordinate in R^{3}. Let E_{1},E_{2},E_{3} be three linearly independent vector fields in R^{3} which satisfies
Let g be the Riemannian metric defined by
Let η be the 1form defined by η(W)=g(W,E_{1}) for any W∈TM. Let φ be the (1,1) tensor field defined by
Then using the linearity of η and g we have
for any U,W∈TM. Now for E_{1}=ξ, the structure (φ,ξ,η,g) defines an almost contact metric structure on M. Using the Koszula formula for the Riemannian metric g, we can straightforwardly calculate
From the above, it follows that the manifold under consideration is a Sasakian manifold of dimension 3. Using the above relations, we can straightforwardly calculate the nonvanishing components of the curvature tensor R as follows:
and the components which can be obtained from these by the symmetry properties. From the above, we can simply calculate the nonvanishing components of the Ricci tensor S and Ricci operator Q as follows:
Since {E_{1},E_{2},E_{3}} forms a basis of the threedimensional Sasakian manifold, any vector field X,Y,Z∈TM can be written as
where a_{i},b_{i},c_{i}∈R^{+} (the set of all positive real numbers), i=1,2,3. Now
and
In view of (38), we have the following;
where
and
Let us now consider the components of the 1forms as
where v_{2}u_{1}−v_{1}u_{2}≠0. From (15), we have
for i=1,2,3. In view of (38), (39), (43) and (44), it can be easily shown that the manifold satisfies the relation (45). Hence the manifold under consideration is a hyper generalized φrecurrent Sasakian manifold, which is not φrecurrent. This leads to the following;
Theorem 4
There exists a threedimensional hyper generalized φrecurrent Sasakian manifold, which is neither φsymmetric nor φrecurrent.
Quasi generalized φrecurrent Sasakian manifold
In the paper [19], the authors studied quasi generalized recurrent manifolds and obtain some interesting results. Motivated by this work, we extend the notion called quasi generalized φrecurrent manifolds. In this section, we study quasi generalized φrecurrent Sasakian manifolds.
Definition 3
A Sasakian manifold M is said to be quasi generalized φrecurrent manifold if its curvature tensor R satisfies the condition
for all X,Y,Z∈TM, where C and D are two nonvanishing 1forms such that C(X)=g(X,μ_{1}), D(X)=g(X,μ_{2}) and the tensor F is defined by
for all X,Y,Z∈TM. Here μ_{1} and μ_{2} are vector fields associated with 1forms C and D respectively. Especially, if the 1form D vanishes, then (46) turns into the notion of φrecurrent manifold.
Note: In view of (46) and (47), we say that locally quasi generalized φrecurrent Sasakian manifold is a locally generalized φrecurrent manifold.
We begin this section with the following:
Theorem 5
A quasi generalized φrecurrent Sasakian manifold M is an Einstein manifold and moreover the associated vector fields μ_{1} and μ_{2} of the 1forms C and D respectively are codirectional.
Proof
Using the same steps as in the proof of Theorem 1, we get the relation
Again using the same steps as in the Theorem 2, we get the equations
for all Y,W. This completes the proof of the theorem. □
Equation (48) leads to the following:
Theorem 6
A quasi generalized φrecurrent Sasakian manifold is a super generalized Riccirecurrent manifold.
From (48), it follows that
This leads to the following:
Theorem 7
In a quasi generalized φrecurrent Sasakian manifold, the 1forms C and D are related by the Eq. (51).
Corollary 7.1
In a quasi generalized φrecurrent Sasakian manifold with nonzero constant scalar curvature, the associated 1forms C and D are related by
Now suppose that quasi generalized φrecurrent Sasakian manifold is quasi generalized Riccirecurrent [19]. Then from (48) we have 2n+1=2n−1, which is not possible. Therefore we can state the following;
Theorem 8
A quasi generalized φrecurrent Sasakian manifold can not be a quasi generalized Riccirecurrent manifold.
In view of (46) and (6), we obtain
From (52) and second Bianchi identity we get
Contracting the above relation over Y and Z and using (47), we get
Again contracting (54) over X and U and using (50), we get
where \(\beta =\frac {r}{2}+\frac {2n\left (2n^{2}1\right)}{14n^{2}}\) and \(\gamma =\frac {2n(14n)}{14n^{2}}\). Hence we can state the following;
Theorem 9
In a quasi generalized φrecurrent Sasakian manifold, the Ricci tensor S and vector field μ_{2} are related by the Eq. (55).
Definition 4
[25] Let M be an almost contact metric manifold with Ricci tensor S. The ∗Ricci tensor and ∗scalar curvature of M are defined repectively by
Definition 5
[26] An almost contact metric manifold M is said to be weakly φEinstein if
for some function β. Here S^{φ} denotes the symmetric part of S^{∗}, that is,
we call S^{φ}, the φRicci tensor on M and the symmetric tensor g^{φ} is defined by g^{φ}(X,Y)=g(φX,φY). When β is constant, M is said to be φEinstein.
In a Sasakian manifold we know the following relation
Using (57) and the relation g((∇_{W}R)(X,Y)Z,ξ)=−g((∇_{W}R)(X,Y)ξ,Z) in (52), we have
from which it follows that
Replacing Z by φZ in the foregoing equation, we obtain
Since g(R(X,Y)φW,U)=g(R(X,Y)W,φU) and g((∇_{W}R)(X,Y)φZ,U)=g((∇_{W}R)(X,Y)Z,φU), using these equation in (60), we get
Contracting (61) over Y and Z and using (47), we get
In view of (48), we have
Substituting U=ξ in (63), we get
From (64) and Definition 5, we conclude that it is φEinstein. Hence we can state the following;
Theorem 10
A quasi generalized φrecurrent Sasakian manifold is an φEinstein manifold.
In view of (7) and (64), we have the following;
Theorem 11
A quasi generalized φrecurrent Sasakian manifold is an ∗ ηEinstein manifold.
Example of a quasi generalized φrecurrent Sasakian manifold
In this section, we give an example of a quasi generalized φrecurrent Sasakian manifold. We take the threedimensional manifold M={(x,y,z)∈R^{3}:z>0}, where (x,y,z) are the standard coordinates in R^{3}. Let E_{1},E_{2},E_{3} be linearly independent global frame on M given by
Let g be the Riemannian metric defined by
Let η be the 1form defined by η(W)=g(W,E_{3}) for any W∈TM. Let φ be the (1,1) tensor field defined by
Then using the linearity of η and g we have
for any U,W∈TM. Then for E_{3}=ξ, the structure (φ,ξ,η,g) defines an almost contact metric structure on M. Let ∇ be the LeviCivita connection with respect to the metric g. Then we have
Using the Koszula formula for the Riemannian metric g, we can easily calculate
From the above, it follows that the manifold under consideration is a Sasakian manifold of 3dimension. Using the above relations, we can easily calculate the nonvanishing components of the curvature tensor R as follows:
and the components which can be obtained from these by the symmetry properties. Since {E_{1},E_{2},E_{3}} forms a basis of the threedimensional Sasakian manifold, any vector field X,Y,Z∈TM can be written as
where a_{i},b_{i},c_{i}∈R^{+} (the set of all positive real numbers), i=1,2,3. Now
and
In view of (65), we have the following:
where
and
Let us now consider the components of the 1forms as
where v_{3}u_{2}+u_{3}v_{2}≠0. From (46), we have
for i=1,2,3. In view of (65), (66), (70), and (71), it can be easily shown that the manifold satisfies the relation (72). Hence, the manifold under consideration is a quasi generalized φrecurrent Sasakian manifold, which is not φrecurrent. This leads to the following:
Theorem 12
There exists a threedimensional quasi generalized φrecurrent Sasakian manifold, which is neither φsymmetric nor φrecurrent.
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Acknowledgements
The author D.M. Naik is financially supported by University Grants Commission, New Delhi (Ref. No.:20/12/2015 (ii)EUV) in the form of Junior Research Fellowship. The authors would like to express their deep thanks to the referee for his/her careful reading and many valuable suggestions towards the improvement of the paper.
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Venkatesha, V., Kumara, H. & Naik, D. On a class of generalized φrecurrent Sasakian manifold. J Egypt Math Soc 27, 19 (2019). https://doi.org/10.1186/s4278701900220
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Keywords
 Hyper generalized recurrent
 Quasi generalized recurrent
 Generalized φrecurrent
 Generalized recurrent
 Riccirecurrent
AMS Subject Classification
 Primary 53C25
 53C15
 Secondary 53D15