Abstract
The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms.
1. Introduction
Alegre et al. [1] introduced and studied the generalized Sasakian-space-forms. The authors Alegre and Carriazo [2], Somashekhara and Nagaraja [3, 4], and De and Sarkar [5, 6] studied the generalized Sasakian-space-forms. An almost contact metric manifold is said to be a generalized Sasakian-space-form if there exist differentiable functions such that curvature tensor of is given by for any vector fields , , on , where In this paper, we study the curvature properties like flatness, symmetry, and semisymmetry properties in a generalized Sasakian-space-form by considering a pseudoprojective curvature tensor.
The paper is organized as follows. Section 2 of this paper contains some preliminary results on the generalized Sasakian-space-forms. In Section 3, we study the pseudoprojectively flat generalized Sasakian-space-form and obtain necessary and sufficient conditions for a generalized Sasakian-space-form to be pseudoprojectively flat. In the next section, we deal with pseudoprojectively semisymmetric generalized Sasakian-space-forms, and it is proved that a generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is pseudoprojectively flat and . The last section is devoted to the study of -flat and -semi symmetric generalized Sasakian-space-forms. In this section, we prove that the associated functions are linearly dependent.
In a -dimensional almost contact metric manifold, the pseudoprojective curvature tensor [7] is defined by where and are constants and , , and are the Riemannian curvature tensor of type , the Ricci tensor, and the scalar curvature of the manifold, respectively. If , , then (1.3) takes the form where is the projective curvature tensor. A manifold shall be called pseudoprojectively flat if the pseudoprojective curvature tensor . It is known that the pseudoprojectively flat manifold is either projectively flat (if ) or Einstein (if and ).
2. Preliminaries
A -dimensional -differentiable manifold is said to admit an almost contact metric structure if it satisfies the following relations: where is a tensor field of type , is a vector field, is a 1-form, and is a Riemannian metric on . A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies for all vector fields and .
In a generalized Sasakian-space-form, the following hold:
3. Pseudoprojectively Flat Generalized Sasakian-Space-Forms
If the generalized Sasakian-space-form under consideration is pseudoprojectively flat, then from (1.3) we have where and are constants and .
Now taking in (3.1) and using (2.1), (2.2), (2.7), and (2.9), we get Again putting in (3.2), we get The aforementioned equation implies That is, either or If , and , then, from (1.3), it follows that . Thus in this case pseudoprojective flatness and projective flatness are equivalent.
If , and , then comparing (2.10) and (3.6), we get Using (3.7) in (2.9), we get Let be an orthonormal basis of the tangent space at each point of the manifold. Taking and summing over , we obtain This shows that is Einstein with a scalar curvature . Thus we state the following.
Theorem 3.1. A pseudoprojectively flat generalized Sasakian-space-form is either projectively flat or an Einstein manifold with a scalar curvature .
Suppose that (3.7) holds. Then in view of (2.7) and (2.9), we can write (1.3) as where Replacing by and by , we get Let be an orthonormal basis of the tangent space at each point of the manifold.
Taking and summation over , , we get Again putting = = and taking summation over , , we get with . In view of (3.7), we get .
Now (2.7) reduces to the form from which we have , and consequently By using (3.14) and (3.15) in (1.3), we get . This leads to the following.
Theorem 3.2. A -dimensional generalized Sasakian-space-form is pseudoprojectively flat if and only if , , and .
Alegre and Carriazo [2] proved that any contact metric generalized Sasakian-space-form with a dimension greater than or equal to five is a Sasakian manifold and , , and must be constants.
Thus from (3.14), we have the following theorem.
Theorem 3.3. A -dimensional generalized Sasakian-space-form with a dimension greater than or equal to 5 is of constant curvature if and only if , , , and .
4. Pseudoprojective Semisymmetric Generalized Sasakian-Space-Form
Definition 4.1. If a generalized Sasakian-space-form satisfies then the manifold is said to be pseudoprojectively semisymmetric manifold.
By using (1.3), (2.1), (2.2), (2.7), and (2.9), we have Taking in (4.2), we get Again putting in (4.2), we get From (4.1), we have Taking and contracting the above with respect to , we get Putting in (4.6) and with the help of (4.2) and (4.3), we get either or Let be an orthonormal basis of the tangent space at each point of the manifold of the manifold. Putting and taking summation over , , and using (4.2) and (4.4), we obtain where Now contracting (4.9), we obtain Using (4.10) in (4.11), we get In view of (2.10), (4.12) yields From (2.9) and (4.13), we have Now using (4.12) and (4.14) in (4.2), we get Plugging (4.15) in (4.6), we obtain Therefore by taking into account (4.7) and (4.16), we have either or is pseudoprojectively flat.
Conversely, suppose that . Then, from (2.1), (2.2) and (2.7), we have . Hence . If the space-form is pseudoprojectively flat then clearly it is pseudoprojectively semisymmetric. Hence we can state the following.
Theorem 4.2. A -dimensional generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is either pseudoprojectively flat or .
By combining Theorems 3.2 and 4.2, we have the following.
Corollary 4.3. A -dimensional generalized Sasakian-space-form is pseudoprojectively flat if and only if or and .
5. -Curvature Tensor in a Generalized Sasakian-Space-Form
In a -dimensional Riemannian manifold , the -curvature tensor is given by [8] where are some smooth functions on . For different values of , the -curvature tensor reduces to the curvature tensor, quasiconformal curvature tensor, conformal curvature tensor, conharmonic curvature tensor, concircular curvature tensor, pseudoprojective curvature tensor, projective curvature tensor, -projective curvature tensor, -curvature tensors , and -curvature tensors .
Suppose that is -flat. Then from (5.1), we have In view of (2.7), (2.8), and (2.9) in (5.2), we have Putting in (5.3), we get If we choose a unit vector orthogonal to and taking , then making use of (2.1) and (2.3) in (5.4), we obtain Putting in (5.5), we have where Thus we have the following.
Theorem 5.1. If a -dimensional generalized Sasakian-space-form is -flat, then (5.6) holds.
From the above theorem, we discuss the following cases.
Case . If is quasiconformally flat, then , , . Putting these in (5.7), we obtain .
If is conharmonically flat, then , , , . Putting these in (5.7), we get .
Similarly for -flat, -flat, -flat, -flat spaces, (5.7) gives .
Case . If is conformally flat, then , , , .
Putting these in (5.7), we obtain . Hence .
Case . If is pseudoprojectively flat, then ,.
By putting these values in (5.7), we have .
If is projectively flat, then .
Making use of the above functional values in (5.7), we get .
Similarly for concircularly flat, -projectively flat, -flat, -flat, -flat, -flat, and -flat spaces, (5.7) gives .
Case . If is -flat, then .
Putting these in (5.7), we obtain that .
If is -flat, then . Putting these in (5.7), we have .
Similarly, for a -flat space, (5.7) gives .
Summarizing the above cases, we have the following corollaries.
Corollary 5.2. If a -dimensional generalized Sasakian-space-form is either quasiconformally flat, conharmonically flat, -flat, -flat, -flat, or -flat, then ,, and are linearly dependent.
Corollary 5.3. If a -dimensional generalized Sasakian-space-form is conformally flat, then .
The above corollary was already proved by Kim [9] and Sarkar and De [10].
Corollary 5.4. If a -dimensional generalized Sasakian-space-form is either pseudoprojectively flat, projectively flat, concircularly flat, -projectively flat, -flat, -flat, -flat, -flat, or -flat, then and are linearly dependent.
Corollary 5.5. If a -dimensional generalized Sasakian-space-form is either -flat, -flat, or -flat, then and are linearly dependent.
5.1. -Semisymmetric Generalized Sasakian-Space-Form
Definition 5.6. is -semisymmetric if holds in .
We know that From (5.8) and (5.9), we have By using (5.1) in (5.10), we have Let be an orthonormal basis of the tangent space at each point of the manifold. Contracting (5.11) with respect to and putting , also taking summation over , , and making use of (2.1), (2.4), (2.7), (2.9), and (2.8), we have where Changing to in (5.12) and also in view of (2.1) and (2.2), (2.9) yields Thus we can state the following.
Theorem 5.7. A -semisymmetric generalized Sasakian-space-form is -Einstein provided .