Abstract
The main aim of this study is to investigate the effects of the curvature flatness, divergence-free characteristic, and symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the curvature flat warped manifold are Einstein manifold. Besides that, the forms of the curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, symmetric warped product manifold is considered.
1. Introduction
Curvature tensors play a significant role in mathematics and physics. This is why many researchers have introduced and studied many curvature tensors in various ways, as well as they have shown the importance of these curvature tensors. For instance, the deviation of a space from constant curvature is measured by the concircular curvature tensor (for more details, see [1]). The Weyl curvature tensor describes the distorting but volume-preserving tidal effects of gravitation on a material body.
The curvature tensor was first coined by De et al. in 2021 [2]. This curvature tensor is a good generalization of projective [3], conharmonic [4], projective [5], and the set of curvature tensors which was introduced by Pokhariyal and Mishra [6–10]. This curvature tensor is given bywhere are constants, is the Riemann tensor, and is the Ricci tensor [2]. The authors studied this curvature tensor on pseudo-Riemannian manifolds and space times of general relativity. It is proved that pseudo-Riemannian manifolds will be Einstein manifold if admits a traceless curvature tensor and will be of constant scalar curvature if is of curvature flat. Pseudo-Riemannian manifolds with divergence-free characteristic were investigated in Gray’s seven subspaces. As a final point, they studied perfect fluid space times when the curvature tensor is flat, and in this case, many interesting results are obtained.
Geometers have considered all well-known curvature tensors on the warped product manifolds. For instance, curvature tensor on warped product manifolds is studied in [11]. Also, concircular curvature tensor on warped product manifold is considered in [12]. Motivated by these kinds of studies and others, this paper aims to investigate the curvature tensor on the warped product manifolds.
This paper is organized as follows: In Section 2, the basic properties of the warped product manifold are presented. In Section 3, we consider the curvature flat warped product manifold. We prove that the base and the fiber manifolds of the curvature flat warped product manifold are Einstein manifold; also in this case, the forms of the curvature tensor on the base and the fiber manifolds are obtained. Section 4 is devoted to study the divergence-free warped product manifold. It is proven that the base and the fiber manifolds of the warped product manifold with divergence-free characteristic are of constant scalar curvature. In addition, these factor manifolds of the divergence-free warped product whose Ricci tensor is of Codazzi type are Ricci symmetric manifolds. Finally, we prove that the warped product manifold is symmetric if and only if the base and the fiber manifolds are symmetric manifolds.
2. On Singly Warped Product Manifold
Let and be two pseudo-Riemannian manifolds with dimensions and , where . And, let be a smooth positive function on . Consider the product manifold with its natural projections and . Then, the singly warped product manifold is the product manifold furnished with the metric tensor
The manifold is called the base manifold, whereas is called the fiber manifold [13, 14]. A warped product manifold is called trivial if the warping function is constant. In this case, is the Riemannian product , where is the manifold equipped with metric , which is homothetic to .
Curvatures of the warped product manifold depend on the curvatures of its fiber and base manifolds. It is noted that the curvatures of the Riemannian product manifold split as a sum of the corresponding curvatures of the first and second factor manifolds since both of the metric and the Levi–Civita connection split as a sum. It is natural now to discuss the deviation in the relation between the different curvature formulas in warped product manifolds and their factor manifolds due to the existence of a nontrivial warping function.
Let , denote the basis vector fields on a neighborhood of the base manifold , where , whereas , denote the basis vector fields on a neighborhood of the fiber manifold , where . Likewise, , denote the basis vector fields on a neighborhood of the warped product manifold , where . The local components of the metric tensor of the warped product manifold are
The local components of the Levi–Civita connection on the warped product are as follows:where and .
On the warped product , the local components of the Riemannian curvature tensor are given by [15–18]where and is a tensor of type with local components and .
The local components of the Ricci curvature of the warped product are the following [16, 17]:where .
It is well known that [16, 17]where “semicolon” refers to the covariant derivative with respect to the metric.
Also,
3. Curvature Flat Singly Warped Product Manifolds
In this section, we consider that the warped product manifold is a curvature flat manifold. The local components of the considered curvature tensor of the warped product manifold , which in general do not vanish identically, are the following , , , and , whereas the local components and vanish.
Let us calculate the first component of the curvature tensor of the warped product manifold which is
In virtue of (3) and (9), we have
Utilizing equations (3), (7), (8), and (10) in equation (14), we infer
Suppose that is curvature flat, that is, . Thus,
A contraction with implies
One more contraction with gives
Equation (17) and (18) together implywhich means that the base manifold is Einstein manifold.
Contracting equation (16) with gives
Multiplying equation (20) with implies
Substituting equation (21) into (20), we havewhich means that the fiber manifold is Einstein manifold.
The second component of the curvature tensor is
In virtue of (3) and (9), one gets
The use of equations (3), (7), (8), and (10) implies
Now, consider that curvature tensor is flat; that is, , and hence,
A contraction with implies
Again, contracting equation (27) with gives
Combining the previous two equations, we reveal that
Similarly, we can obtain
From the above discussion, we are in a position to state the following.
Theorem 1. Let be a curvature flat singly warped product manifold furnished with the metric tensor . Then, the base and the fiber manifolds of the warped product manifold are Einstein manifold.
Moving on to the next component, we have
Using (3), (8), and (5), we obtain
The previous equation can be rewritten in the following form:
Remember that the curvature tensor on the base manifold is of the form
Consequently, we can obtain the following:
Suppose that is a curvature flat; that is, . This leads towhich is the form of the curvature tensor of the base manifold . Thus, we can state the following theorem.
Theorem 2. Let be a curvature flat singly warped product manifold equipped with the metric tensor . Then, the curvature tensor on the base manifold is given by
Assume that ; then, equation (36) implieswhich means that the base manifold is curvature flat.
Corollary 1. The base manifold of the warped product manifold is curvature flat if the warped product manifold is curvature flat and .
The last component of the curvature tensor is
The previous equation can be rewritten in the following form:
The curvature tensor on the fiber manifold is given by
Thus,
If is curvature flat, that is, , then
Thus, we have the following.
Theorem 3. Let be a curvature flat singly warped product manifold with the metric tensor . Then, the curvature tensor on the fiber manifold is of the form
4. Divergence-Free Warped Product Manifold
The divergence of the curvature tensor is given by [2]
If curvature tensor is divergence-free, that is, , then
Contracting with and using the relation , we get
If , then . And hence, equation (46) reduces to
We thus have the following.
Lemma 1. A warped manifold with divergence-free curvature tensor is of constant scalar curvature and Ricci tensor satisfies equation (49), provided .
The divergence component of the curvature tensor on the warped product manifold is
If is divergence-free, that is, , then
Using the obtained result in the previous lemma, we can have
In view of equation (12), we infer
Contracting with and using , we get
If , thenwhich means that the fiber manifold of the warped product manifold is of constant scalar curvature. Thus, we can state the following theorem.
Theorem 4. The fiber manifold of divergence-free warped product manifold is of constant scalar curvature.
The next divergence component is
Assuming that is divergence-free and utilizing the obtained result in the previous lemma, we have
In virtue of equation (12), we get
If , then
Multiplying this with and using , we have
If , thenwhich means that the base manifold of the warped product manifold is of constant scalar curvature. Thus, we conclude the following.
Theorem 5. The base manifold of divergence-free warped product manifold is of constant scalar curvature, provided .
Now, consider the warped product has a Codazzi Ricci tensor; that is, . And consequently, is of constant scalar curvature. Thus, equation (46) leads to
Proposition 1. A warped product manifold with Codazzi Ricci tensor is divergence-free if and only if it has symmetric Ricc'i tensor, provided .
Now, the divergence component of the curvature tensor is
Assume that the warped product manifold is divergence-free, and hence, one gets
Using equation (12), we get
If , then
Thus, we have the folllowing.
Theorem 6. Let be a divergence-free warped product whose Ricci tensor is of Codazzi type. Then, the Ricci tensor of the fiber manifold is symmetric.
Also, the divergence component is
If is divergence-free, then
In view of equation (12), we get
If and , then
We thus can state the following.
Theorem 7. Let be a warped product with Codazzi Ricci tensor. Then the Ricci tensor of the base manifold is symmetric.
5. Semisymmetries of the Curvature Tensor
It is well known that a manifold is said to be semisymmetric if its Riemann tensor satisfies
A manifold is said to be Ricci semisymmetric if its Ricci tensor satisfies
The curvature tensor is called semisymmetric if
Applying on both sides of equation (1), one can have
Thus, we have the following:
Proposition 2. A pseudo-Riemannian manifold admits a semisymmetric curvature tensor if and only if is semisymmetric.
Now, assume that has a semisymmetric curvature tensor; that is,
Thus, we have
A contraction with implies
A multiplication with implies
If , then
And hence, equation (77) becomes
If , we havewhich means that is Ricci semisymmetric.
Proposition 3. A pseudo-Riemannian manifold with semisymmetric curvature tensor is Ricci semisymmetric.
Taking the covariant derivative of the first component of the curvature tensor, which is given by equation (14), we infer
Using equations (11) and (12) in the previous equation, we have
Suppose that the curvature tensor is symmetric; that is, . Thus,
A contraction with gives
If , one may havewhich means that the fiber manifold of the warped product manifold is Ricci symmetric. We thus can state the following.
Theorem 8. Let be a warped product manifold with symmetric curvature tensor. Then, the fiber manifold of is a Ricci symmetric manifold.
The covariant derivative of the component is
The use of equations (3), (11), and (12) implies
Thus, we can state the following.
Theorem 9. Let be a warped product manifold with symmetric curvature tensor. Then, the fiber manifold has symmetric curvature tensor.
The covariant derivative of the component is
Utilizing (3), (11), and (12) entails that
If , we have
We thus can state the following theorem.
Theorem 10. Let be a warped product manifold with symmetric curvature tensor. Then, the base manifold has a symmetric curvature tensor, provided .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was supported by the Researchers Supporting Project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia.