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Index of Quasiconformally Symmetric Semi-Riemannian Manifolds
We find the index of -quasiconformally symmetric and -concircularly symmetric semi-Riemannian manifolds, where is metric connection.
In 1923, Eisenhart  gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy  proved that a second-order parallel symmetric nonsingular tensor in a real-space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, Sharma  proved that any second-order parallel tensor (not necessarily symmetric) in a real-space form of dimension greater than is proportional to the Riemannian metric. In 1939, Thomas  defined and studied the index of a Riemannian manifold. A set of metric tensors (a metric tensor on a differentiable manifold is a symmetric nondegenerate parallel tensor field on the differentiable manifold) is said to be linearly independent if implies that The set is said to be a complete set if any metric tensor can be written as More precisely, the number of linearly independent metric tensors in a complete set of metric tensors of a Riemannian manifold is called the index of the Riemannian manifold [4, page 413]. Thus, the problem of existence of a second-order parallel symmetric tensor is closely related with the index of Riemannian manifolds. Later, in 1968, Levine and Katzin  studied the index of conformally flat Riemannian manifolds. They proved that the index of an -dimensional conformally flat manifold is or according as it is a flat manifold or a manifold of nonzero constant curvature. In 1981, Stavre  proved that if the index of an -dimensional conformally symmetric Riemannian manifold (except the four cases of being conformally flat, of constant curvature, an Einstein manifold or with covariant constant Einstein tensor) is greater than one, then it must be between and . In 1982, Starve and Smaranda  found the index of a conformally symmetric Riemannian manifolds with respect to a semisymmetric metric connection of Yano . More precisely, they proved the following result: "Let a Riemannian manifold be conformally symmetric with respect to a semisymmetric metric connection . Then (a) the index is if there is a vector field such that and , where and are the Einstein tensor field and the scalar curvature with respect to the connection , respectively; and (b) the index satisfies if ."
A real-space form is always conformally flat, and a conformally flat manifold is always conformally symmetric. But the converse is not true in both the cases. On the other hand, the quasiconformal curvature tensor  is a generalization of the Weyl conformal curvature tensor and the concircular curvature tensor. The Levi-Civita connection and semisymmetric metric connection are the particular cases of a metric connection. Also, a metric connection is Levi-Civita connection when its torsion is zero and it becomes the Hayden connection  when it has nonzero torsion. Thus, metric connections include both the Levi-Civita connections and the Hayden connections (in particular, semisymmetric metric connections).
Motivated by these circumstances, it becomes necessary to study the index of quasiconformally symmetric semi-Riemannian manifolds with respect to any metric connection. The paper is organized as follows. In Section 2, we give the definition of the index of a semi-Riemannian manifold and give the definition and some examples of the Ricci symmetric metric connections . In Section 3, we give the definition of the quasiconformal curvature tensor with respect to a metric connection . We also obtain a complete classification of -quasiconformally flat (and in particular, quasiconformally flat) manifolds. In Section 4, we find out the index of -quasiconformally symmetric manifolds and -concircularly symmetric manifolds. In the last section, we discuss some of applications in theory of relativity.
2. Index of a Semi-Riemannian Manifold
Let be an -dimensional differentiable manifold. Let be a linear connection in . Then torsion tensor and curvature tensor of are given by for all , where is the Lie algebra of vector fields in . By a semi-Riemannian metric  on , we understand a nondegenerate symmetric tensor field . In , a semi-Riemannian metric is called simply a metric tensor. A positive definite symmetric tensor field is well known as a Riemannian metric, which, in , is called a fundamental metric tensor. A symmetric tensor field of rank less than is called a degenerate metric tensor .
Let be an -dimensional semi-Riemannian manifold. A linear connection in is called a metric connection with respect to the semi-Riemannian metric if . If the torsion tensor of the metric connection is zero, then it becomes Levi-Civita connection , which is unique by the fundamental theorem of Riemannian geometry. If the torsion tensor of the metric connection is not zero, then it is called a Hayden connection [10, 12]. Semisymmetric metric connections  and quarter symmetric metric connections  are some well-known examples of Hayden connections.
Let be an -dimensional semi-Riemannian manifold. For a metric connection in , the curvature tensor with respect to the satisfies the following condition: for all , where The Ricci tensor and the scalar curvature of the semi-Riemannian manifold with respect to the metric connection is defined by where is any orthonormal basis of vector fields in the manifold and . The Ricci operator with respect to the metric connection is defined by Define Then, consider The tensor is called tensor of Einstein  with respect to the metric connection . If is symmetric, then is also symmetric.
Definition 2.1. A metric connection with symmetric Ricci tensor will be called a “Ricci-symmetric metric connection.”
Example 2.2. In a semi-Riemannian manifold , a semisymmetric metric connection of Yano  is given by where is Levi-Civita connection, is a vector field, and is its associated form given by . The Ricci tensor with respect to is given by where is the Ricci tensor, and is a tensor field defined by The Ricci tensor is symmetric if form, is closed.
Example 2.3. An -almost para contact metric manifold is given by where is a tensor field of type , is form, is a vector field and . An -almost para contact metric manifold satisfying is called an -para Sasakian manifold . In an -para Sasakian manifold, the semisymmetric metric connection given by is a Ricci symmetric metric connection.
Example 2.4. An almost contact metric manifold is given by where is a tensor field of type , is -form and is a vector field. An almost contact metric manifold is a Kenmotsu manifold  if and is a Sasakian manifold  if In an almost contact metric manifold , the semisymmetric metric connection given by is a Ricci symmetric metric connection if is Kenmotsu, but the connection fails to be Ricci symmetric if is Sasakian.
Let be an -dimensional semi-Riemannian manifold equipped with a metric connection . A symmetric tensor field , which is covariantly constant with respect to , is called a special quadratic first integral (for brevity SQFI)  with respect to . The semi-Riemannian metric is always an SQFI. A set of SQFI tensors with respect to is said to be linearly independent if implies that The set is said to be a complete set if any SQFI tensor with respect to can be written as The “index”  of the manifold with respect to , denoted by , is defined to be the number of members in a complete set .
We will need the following Lemma.
Lemma 2.5. Let be an -dimensional semi-Riemannian manifold equipped with a Ricci symmetric metric connection . Then the following statements are true.(a)If , then . Conversely, if is constant and then .(b)If and is a nonvanishing differentiable function such that and are linearly dependent, then .
The proof is similar to Lemmas 1.2 and 1.3 in  for a semisymmetric metric connection and is therefore omitted.
3. Quasiconformal Curvature Tensor
Let be an -dimensional semi-Riemannian manifold equipped with a metric connection . The conformal curvature tensor with respect to the is defined by [19, page 90] as follow: and the concircular curvature tensor with respect to is defined by (, [21, page 87]) as follows: As a generalization of the notion of conformal curvature tensor and concircular curvature tensor, the quasiconformal curvature tensor with respect to is defined by  as follows: where and are constants. In fact, we have Since there is no restrictions for manifolds if and , therefore it is essential for us to consider the case of or . From (3.4) it is clear that if and , then ; if and , then .
Now, we need the following.
Definition 3.1. A semi-Riemannian manifold equipped with a metric connection is said to be(a)-quasiconformally flat if ,(b)-conformally flat if , and (c)-concircularly flat if . In particular, with respect to the Levi-Civita connection , -quasiconformally flat, conformally flat, and -concircularly flat become simply quasiconformally flat, conformally flat, and concircularly flat, respectively.
Definition 3.2. A semi-Riemannian manifold equipped with a metric connection is said to be(a)-quasiconformally symmetric if ,(b)-conformally symmetric if , and(c)-concircularly symmetric if . In particular, with respect to the Levi-Civita connection , -quasiconformally symmetric, -conformally symmetric, and -concircularly symmetric become simply quasiconformally symmetric, conformally symmetric, and concircularly symmetric, respectively.
Theorem 3.3. Let be a semi-Riemannian manifold of dimension . Then is -quasiconformally flat if and only if one of the following statements is true:(i), , and is -conformally flat,(ii), , is -conformally flat, and -concircularly flat,(iii), and Ricci tensor with respect to satisfies
where is the scalar curvature with respect to .
Proof. Using in (3.3), we get
from which we obtain the following:
Case 1 ( and ). Then from (3.3) and (3.1), it follows that , which gives . This gives the statement (i).
Case 2 ( and ). Then from (3.7) Using (3.8) in (3.6), we get Since , then by (3.2) and by using (3.9), (3.8) in (3.1), we get . This gives the statement (ii).
Case 3 ( and , we get (3.5)). This gives the statement (iii). Converse is true in all cases.
Corollary 3.4 (see , Theorem 5.1). Let be a semi-Riemannian manifold of dimension . Then is quasiconformally flat if and only if one of the following statements is true:(i), , and is conformally flat,(ii), ,and is of constant curvature, and(iii), , and is Einstein manifold.
Remark 3.5. In , the following three results are known.(a)[23, Proposition 1.1]. A quasiconformally flat manifold is either conformally flat or Einstein.(b)[23, Corollary 1.1]. A quasiconformally flat manifold is conformally flat if the constant .(c)[23, Corollary 1.2]. A quasiconformally flat manifold is Einstein if the constants and .
However, the converses need not be true in these three results. But, in Corollary 3.4 we get a complete classification of quasiconformally flat manifolds.
4. -Quasiconformally Symmetric Manifolds
Let be an -dimensional semi-Riemannian manifold equipped with the metric connection . Let be the curvature tensor of with respect to the metric connection . If is a parallel symmetric tensor with respect to the metric connection , then we easily obtain that The solutions of (4.1) is closely related to the index of quasiconformally symmetric and concircularly symmetric manifold with respect to the .
Lemma 4.1. Let be an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and . Then
Theorem 4.2. If is an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and , then (4.1) takes the form If , then (4.6) has the general solution where is an arbitrary nonvanishing differentiable function.
Proof. Using (4.4) in (4.1), we get Let be an orthonormal basis of vector fields in . Taking in (4.8) and summing up to terms, then, using (4.2), we have Interchanging and in (4.9) and subtracting the so-obtained formula from (4.9), we deduce that Now, interchanging and , , and in (4.8) and taking the sum of the resulting equation and (4.8) and using (4.9) and (4.10), we get (4.6). If , then using (2.7) leads to (4.7).
Theorem 4.3. If is an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and , and if there is a vector field so that then the solution of (4.1) is , where is a differentiable nonvanishing function.
Theorem 4.4. Let be an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and . If there is a vector field satisfying the condition (4.11), then .
Proof. By Theorem 4.3 and from the fact that and , it follows that is constant. Thus, .
Theorem 4.5. Let be an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and , for which the tensor field is not covariantly constant with respect to the Ricci symmetric metric connection . If , then there is a vector field , so that the equation has the fundamental solutions where is a differentiable nonvanishing function.
Proof. Given that , there is so that the tensorial equation (4.1) has general solution which depends on . is obviously a solution of (4.14) because , also satisfies the tensorial equation (4.1), and given by (4.7) is also a solution of (4.14). Equation (4.14) has at least two solution as . These two solutions are independent. By Lemma 2.5(b) and are independent and we get two fundamental solution of which is , where is a differentiable nonvanishing function.
Theorem 4.6. Let be an -dimensional semi-Riemannian -quasiconformally symmetric manifold, and , for which the tensor field is not covariantly constant with respect to the metric connection . Then .
Proof. Let , be independent vector fields, for which and let and be the fundamental solutions of . Obviously , as are independent. Therefore, we have solutions. This completes the proof.
Remark 4.7. The previous results of this section will be true for -conformally symmetric semi-Riemannian manifold, where is any Ricci symmetric metric connection.
Theorem 4.8. If be an -dimensional semi-Riemannian -concircularly symmetric manifold, then the (4.1) takes the form
Proof. Taking covariant derivative of (3.2) and using , we get which, when used in (4.1), yields Now, we interchange with and with in (4.19) and take the sum of the resulting equation and (4.19), and we get (4.17).
Theorem 4.9. Let be an -dimensional semi-Riemannian -concircularly symmetric manifold. Then .
Proof. By Theorem 4.8 and from the fact that and , we get .
A semi-Riemannian manifold is said to be decomposable  (or locally reducible) if there always exists a local coordinate system so that its metric takes the form where are functions of and are functions of . A semi-Riemannian manifold is said to be reducible if it is isometric to the product of two or more semi-Riemannian manifolds; otherwise, it is said to be irreducible . A reducible semi-Riemannian manifold is always decomposable but the converse needs not to be true.
The concept of the index of a (semi-)Riemannian manifold gives a striking tool to decide the reducibility and decomposability of (semi-)Riemannian manifolds. For example, a Riemannian manifold is decomposable if and only if its index is greater than one . Moreover, a complete Riemannian manifold is reducible if and only if its index is greater than one . A second-order -symmetric parallel tensor is also known as a special Killing tensor of order two. Thus, a Riemannian manifold admits a special Killing tensor other than the Riemannian metric if and only if the manifold is reducible , that is the index of the manifold is greater than . In 1951, Patterson  found a similar result for semi-Riemannian manifolds. In fact, he proved that a semi-Riemannian manifold admitting a special Killing tensor , other than , is reducible if the matrix has at least two distinct characteristic roots at every point of the manifold. In this case, the index of the manifold is again greater than .
By Theorem 4.6, we conclude that a -quasiconformally symmetric Riemannian manifold (where is any Ricci symmetric metric connection, not necessarily Levi-Civita connection) is decomposable, and it is reducible if the manifold is complete.
It is known that the maximum number of linearly independent Killing tensors of order in a semi-Riemannian manifold is , which is attained if and only if is of constant curvature. The maximum number of linearly independent Killing tensors in a four-dimensional spacetime is , and this number is attained if and only if the spacetime is of constant curvature . But, from Theorem 4.6, we also conclude that the maximum number of linearly independent special Killing tensors in a -dimensional Robertson-Walker spacetime [11, page 341] is .
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