Index of quasi-conformally symmetric semi-Riemannian manifolds

We find the index of $\widetilde{\nabla}$-quasi-conformally symmetric and $\widetilde{\nabla}$-concircularly symmetric semi-Riemannian manifolds, where $\widetilde{\nabla}$ is metric connection.


Introduction
In 1923, L.P. Eisenhart [2] gave the condition for the existence of a second order parallel symmetric tensor in a Riemannian manifold. In 1925, H. Levy [10] proved that a second order parallel symmetric non-singular tensor in a real space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, R. Sharma [14] proved that any second order parallel tensor (not necessarily symmetric) in a real space form of dimension greater than 2 is proportional to the Riemannian metric. In 1939, T.Y. Thomas [18] defined and studied the index of a Riemannian manifold. More precisely, the number of metric tensors (a metric tensor on a differentiable manifold is a symmetric non-degenerate parallel (0, 2) tensor field on the differentiable manifold) in a complete set of metric tensors of a Riemannian manifold is called the index of the Riemannian manifold [18, p. 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, J. Levine and G.H. Katzin [9] studied the index of conformally flat Riemannian manifolds. They proved that the index of an n-dimensional conformally flat manifold is n(n + 1)/2 or 1 according as it is a flat manifold or a manifold of non-zero constant curvature. In 1981, P. Stavre [15] proved that if the index of an n-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 2 and n + 1. In 1982, P. Starve and D. Smaranda [17] found the index of a conformally symmetric Riemannian manifolds with respect to a semi-symmetric metric connection of K. Yano [22]. More precisely, they proved the following result: "Let a Riemannian manifold be conformally symmetric with respect to a semi-symmetric metric connection ∇. Then (a) the index i ∇ is 1 if there is a vector field U such that ∇ U E = 0 and ∇ U r = 0, where E and r are the Einstein tensor field and the scalar curvature with respect to the connection ∇, respectively; and (b) the index i ∇ satisfies 1 < i ∇ ≤ n + 1 if ∇ E = 0".
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 quasi-conformal curvature tensor [25] is a generalization of the Weyl conformal curvature tensor and the concircular curvature tensor. The Levi-Civita connection and semi-symmetric metric connection are the particular cases of the metric connection. Also, a metric connection is Levi-Civita connection when its torsion is zero and it becomes the Hayden connection [6] when it has non-zero torsion. Thus, metric connections include both the Levi-Civita connections and the Hayden connections (in particular, semi-symmetric 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 ∇-quasi-conformally flat (and in particular, quasi-conformally flat) manifolds. In Section 4, we find out the index of ∇-quasi-conformally symmetric manifolds and ∇-concircularly symmetric manifolds. In the last section, we discuss some of applications in theory of relativity.

Index of a semi-Riemannian manifold
Let M be an n-dimensional differentiable manifold. Let ∇ be a linear connection in M . Then torsion tensor T and curvature tensor R of ∇ are given by for all X, Y, Z ∈ X(M ), where X(M ) is the Lie algebra of vector fields in M . By a semi-Riemannian metric [11] on M , we understand a non-degenerate symmetric (0, 2) tensor field g. In [18], a semi-Riemannian metric is called simply a metric tensor. A positive definite symmetric (0, 2) tensor field is well known as a Riemannian metric, which, in [18], is called a fundamental metric tensor. A symmetric (0, 2) tensor field g of rank less than n is called a degenerate metric tensor [18].
Let (M, g) be an n-dimensional semi-Riemannian manifold. A linear connection ∇ in M is called a metric connection with respect to the semi-Riemannian metric g if ∇g = 0. 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 [6,23]. Semi-symmetric metric connections [22] and quarter symmetric metric connections [4] are some well known examples of Hayden connections.
Let (M, g) be an n-dimensional semi-Riemannian manifold. For a metric connection ∇ in M , the curvature tensor R with respect to the ∇ satisfies the following condition 3) The Ricci tensor S and the scalar curvature r of the semi-Riemannian manifold with respect to the metric connection ∇ is defined by where {e 1 , . . . , e n } is any orthonormal basis of vector fields in the manifold M and ε i = g(e i , e i ). The Ricci operator Q with respect to the metric connection ∇ is defined by The (0, 2) tensor E is called tensor of Einstein [16] with respect to the metric connection ∇. If S is symmetric then E is also symmetric.
Definition 1 A metric connection ∇ with symmetric Ricci tensor S will be called a Ricci-symmetric metric connection.
Example 1 In a semi-Riemannian manifold (M, g), a semi-symmetric metric connection ∇ of K. Yano [22] is given by where ∇ is Levi-Civita connection, U is a vector field and u is its associated 1-form given by u(X) = g(X, U ). The Ricci tensor S with respect to ∇ is given by where S is the Ricci tensor and α is a (0, 2) tensor field defined by The Ricci tensor S is symmetric if 1-form u is closed.
Example 3 An almost contact metric manifold (M, ϕ, ξ, η, g) is given by where ϕ is a tensor field of type (1, 1), η is 1-form and ξ is a vector field. An almost contact metric manifold is a Kenmotsu manifold and is a Sasakian manifold [13] if In an almost contact metric manifold M the semi-symmetric metric connection ∇ given by is a Ricci symmetric metric connection if M is Kenmotsu, but the connection fails to be Ricci symmetric if M is Sasakian.
Let (M, g) be an n-dimensional semi-Riemannian manifold equipped with a metric connection ∇. A symmetric (0, 2) tensor field H, which is covariantly constant with respect to ∇, is called a special quadratic first integral (for brevity SQFI) [8] with respect to ∇. The semi-Riemannian metric g is always an SQFI. A set of SQFI tensors {H 1 , . . . , H ℓ } with respect to ∇ is said to be linearly independent if implies that c 1 = · · · = c ℓ = 0.
The set {H 1 , . . . , H ℓ } is said to be a complete set if any SQFI tensor H with respect to ∇ can be written as The index [18] of the manifold M with respect to ∇, denoted by i ∇ , is defined to be the number ℓ of members in a complete set {H 1 , . . . , H ℓ }.
We shall need the following Lemma: Lemma 1 Let (M, g) be an n-dimensional semi-Riemannian manifold equipped with a Ricci symmetric metric connection ∇. Then the following statements are true: If ∇ X S = 0 and ψ is a non-vanishing differentiable function such that ψ ∇ X S and g are linearly dependent, then ∇ X E = 0.
The proof is similar to Lemmas 1.2 and 1.3 in [17] for a semi-symmetric metric connection, and is therefore omitted.

Quasi-conformal curvature tensor
Let (M, g) be an n-dimensional (n > 3) semi-Riemannian manifold equipped with a metric connection ∇. The conformal curvature tensor C with respect to the ∇ is defined by [3, p. 90] As a generalization of the notion of conformal curvature tensor and concircular curvature tensor, the quasi-conformal curvature tensor C * with respect to ∇ is defined by [25] where a and b are constants. In fact, we have In particular, with respect to the Levi-Civita connection ∇, ∇-quasi-conformally symmetric, ∇-conformally symmetric and ∇-concircularly symmetric become simply quasiconformally symmetric, conformally symmetric and concircularly symmetric respectively. (ii) a + (n − 2)b = 0, a = 0, M is ∇-conformally flat and ∇-concircularly flat.
(iii) a + (n − 2)b = 0, a = 0 and Ricci tensor S with respect to ∇ satisfies S − r n g = 0, (3.5) where r is the scalar curvature with respect to ∇.
Proof. Using C * = 0 in (3.3) we get S (Y, Z) = r n g(Y, Z). (3.8) Since a = 0, then by (3.2) Z = 0 and by using (3.9), (3.8) in (3.1), we get C = 0. This gives the statement (ii). However, the converses need not be true in these three results. But, in Corollary 1 we get a complete classification of quasi-conformally flat manifolds.

∇-Quasi-conformally symmetric manifolds
Let (M, g) be an n-dimensional semi-Riemannian manifold equipped with the metric connection ∇. Let R be the curvature tensor of M with respect to the metric connection ∇. If H is a parallel symmetric (0, 2) tensor with respect to the metric connection ∇, then we easily obtain The solutions H of (4.1) is closely related to the index of quasi-conformally symmetric and concircularly symmetric manifold with respect to the ∇.
Theorem 6 If (M, g) is an n-dimensional semi-Riemannian ∇-quasi-conformally symmetric manifold, n > 2 and b = 0, then the equation (4.1) takes the form If ∇ U E = 0, then (4.6) has the general solution where f is an arbitrary non-vanishing differentiable function.
Proof. Using (4.4) in (4.1), we get Let {e 1 , . . . , e n } be an orthonormal basis of vector fields in M . Taking X = Z = e i in (4.8) and summing upto n terms, then using (4.2), we have Interchanging Y and V in (4.9) and subtracting the so obtained formula from (4.9), we deduce that Now, interchanging X and Z, Y and V 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 ∇ U E = 0, then using (2.6) leads to (4.7).
Theorem 7 If (M, g) is an n-dimensional semi-Riemannian ∇-quasi-conformally symmetric manifold, n > 2 and b = 0, and if there is a vector field U so that ∇ U E = 0 and ∇ U r = 0, (4.11) then the solution of a equation (4.1) is H = f g, where f is a differentiable non-vanishing function.
Proof. Using (4.11), (4.8) becomes Interchanging X and Z, Y and V in (4.12) and taking the sum of the resulting equation and (4.12), we get Therefore the tensor fields H and g are proportional.
Proof. By Theorem 7 and from the fact that ∇ U g = 0 and ∇ U H = 0, it follows that f is constant. Thus i ∇ = 1.
Theorem 9 Let (M, g) be an n-dimensional semi-Riemannian ∇-quasi-conformally symmetric manifold, n > 2 and b = 0, for which the tensor field E is not covariantly constant with respect to the Ricci symmetric metric connection ∇. If i ∇ > 1, then there is a vector field U , so that the equation has the fundamental solutions where ψ is a differentiable non-vanishing function.
Proof. Given that ∇ U E = 0, there is U so that the tensorial equation (4.1) has general solution which depends on U . g is obviously a solution of (4.13 ) because ∇ U g = 0, g is also satisfy the tensorial equation (4.1) and H U given by (4.7) is also a solution of (4.13). Equation (4.13 ) has at least two solution as i ∇ > 1. These two solution are independent.By Lemma 1(b) ψ ∇ U S and g are independent and we get two fundamental solution of ∇ U H = 0 which is H 1 = g, H 2 = ψ ∇ U S, where ψ is a differentiable nonvanishing function.
Theorem 10 Let (M, g) be an n-dimensional semi-Riemannian ∇-quasi-conformally symmetric manifold, n > 2 and b = 0, for which the tensor field E is not covariantly constant with respect to the metric connection ∇. Then 1 ≤ i ∇ ≤ n + 1.
Proof. Let U i , i = 1, . . . , p be independent vector fields, for which ∇ Ui E = 0, and let ψ i ∇ Ui S and g be the fundamental solutions of ∇ Ui H = 0. Obviously p < n, as U i are independent. Therefore we have p + 1 solutions. This completes the proof.

Remark 11
The previous results of this section will be true for ∇-conformally symmetric semi-Riemannian manifold, where ∇ is any Ricci symmetric metric connection.
Theorem 12 If (M, g) be an n-dimensional semi-Riemannian ∇-concircularly symmetric manifold, then the equation (4.1) takes the form Proof. Taking covariant derivative of (3.2) and using ∇ U Z = 0, we get which, when used in (4.1), yields Now, we interchange X with Z, and Y with V in (4.16) and take the sum of the resulting equation and (4.16), we get (4.15).
Proof. By Theorem 12 and from the fact that ∇ U g = 0 and ∇ U H = 0, we get i ∇ = 1.

Discussion
A semi-Riemannian manifold is said to be decomposable [18] (or locally reducible) if there always exists a local coordinate system x i so that its metric takes the form where g ab are functions of x 1 , . . . , x r and g αβ are functions of x r+1 , . . . , x n . 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 [18]. A reducible semi-Riemannian manifold is always decomposable but the converse need not 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 [18]. Moreover, a complete Riemannian manifold is reducible if and only if its index is greater than one [18]. A second order (0, 2)-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 g if and only if the manifold is reducible [2], that is the index of the manifold is greater than 1. In 1951, E.M. Patterson [12] found a similar result for semi-Riemannian manifolds. In fact, he proved that a semi-Riemannian manifold (M, g) admitting a special Killing tensor K ij , other than g, is reducible if the matrix (K ij ) 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 1.
By Theorem 10, we conclude that a ∇-quasi-conformally 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 2 in a semi-Riemannian manifold (M n , g) is 1 12 n(n + 1) 2 (n + 2), which is attained if and only if M is of constant curvature. The maximum number of linearly independent Killing tensors in a four dimensional spacetime is 50 and this number is attained if and only if the spacetime is of constant curvature [5]. But, from Theorem 10, we also conclude that the maximum number of linearly independent special Killing tensors in a 4-dimensional Robertson-Walker spacetime [11, p. 341] is 5.