Index of Quasiconformally Symmetric Semi-Riemannian Manifolds

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Volume 2012, Article ID 461383,14pages doi:10.1155/2012/461383

Research Article

Index of Quasiconformally Symmetric Semi-Riemannian Manifolds

Mukut Mani Tripathi,

¹

Punam Gupta,

²

and Jeong-Sik Kim

³

1Department of Mathematics and DST-CIMS, Faculty of Science, Banaras Hindu University, Varanasi 221005, India

2Department of Mathematics, School of Applied Sciences, KIIT University, Odisha, Bhubaneswar 751024, India

3GwangJu Jeil High School, Donlibro, Buk-gu, GwangJu 237 33, Republic of Korea

Correspondence should be addressed to Mukut Mani Tripathi,[email protected] Received 26 March 2012; Accepted 30 May 2012

Academic Editor: Nageswari Shanmugalingam

Copyrightq2012 Mukut Mani Tripathi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We find the index of ∇-quasiconformally symmetric and ∇-concircularly symmetric semi- Riemannian manifolds, where∇is metric connection.

1. Introduction

In 1923, Eisenhart1gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy2proved 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, Sharma3proved that any second-order parallel tensornot necessarily symmetricin a real-space form of dimension greater than 2 is proportional to the Riemannian metric. In 1939, Thomas4defined and studied the index of a Riemannian manifold. A set of metric tensorsa metric tensor on a diﬀerentiable manifold is a symmetric nondegenerate parallel 0,2 tensor field on the diﬀerentiable manifold {H1, . . . , H}is said to be linearly independent if

c₁H₁· · ·cH 0, c₁, . . . , c∈R, 1.1 implies that

c₁· · ·c 0. 1.2

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The set{H1, . . . , H}is said to be a complete set if any metric tensorHcan be written as

Hc1H1· · ·cH, c1, . . . , c∈R. 1.3

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 manifold4, 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 Katzin5studied the index of conformally flat Riemannian manifolds. They proved that the index of an n- dimensional conformally flat manifold isnn1/2 or 1 according as it is a flat manifold or a manifold of nonzero constant curvature. In 1981, Stavre 6 proved that if the index of ann-dimensional conformally symmetric Riemannian manifoldexcept 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 n1. In 1982, Starve and Smaranda7found the index of a conformally symmetric Riemannian manifolds with respect to a semisymmetric metric connection of Yano8. More precisely, they proved the following result: ”Let a Riemannian manifold be conformally symmetric with respect to a semisymmetric metric connection∇. Thenathe indexi_∇is 1 if there is a vector fieldUsuch that∇UE0 and∇Ur /0, whereEandrare the Einstein tensor field and the scalar curvature with respect to the connection∇, respectively; andbthe indexi_∇satisfies 1< i_∇≤ n1 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 quasiconformal curvature tensor9is 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 10 when it has nonzero torsion. Thus, metric connections include both the Levi-Civita connections and the Hayden connectionsin 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. InSection 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

LetMbe ann-dimensional diﬀerentiable manifold. Let∇ be a linear connection inM. Then torsion tensorTand curvature tensorRof∇ are given by

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TX, Y ∇XY −∇YX−X, Y, RX, YZ ∇X∇YZ−∇Y∇XZ−∇_X,YZ

2.1

for all X, Y, Z ∈ XM, where XM is the Lie algebra of vector fields in M. By a semi- Riemannian metric11onM, we understand a nondegenerate symmetric0,2tensor field g. In 4, a semi-Riemannian metric is called simply a metric tensor. A positive definite symmetric0,2tensor field is well known as a Riemannian metric, which, in4, is called a fundamental metric tensor. A symmetric0,2tensor fieldgof rank less thannis called a degenerate metric tensor4.

LetM, gbe ann-dimensional semi-Riemannian manifold. A linear connection∇ in Mis called a metric connection with respect to the semi-Riemannian metricg 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 connection10,12.

Semisymmetric metric connections 8 and quarter symmetric metric connections13are some well-known examples of Hayden connections.

LetM, gbe ann-dimensional semi-Riemannian manifold. For a metric connection∇ inM, the curvature tensorRwith respect to the∇ satisfies the following condition:

RX, Y, Z, V RY, X, Z, V 0, RX, Y, Z, V RX, Y, V, Z 0

2.2

for allX, Y, Z, V ∈XM, where

RX, Y, Z, V g

RX, YZ, V

. 2.3

The Ricci tensorSand the scalar curvaturerof the semi-Riemannian manifold with respect to the metric connection∇ is defined by

SX, Y ⁿ

i1

ε_iRe i, X, Y, e_i,

r ⁿ

i1

εiSe i, ei,

2.4

where {e1, . . . , e_n} is any orthonormal basis of vector fields in the manifold M and ε_i gei, ei. The Ricci operatorQwith respect to the metric connection∇ is defined by

SX, Y g QX, Y

, X, Y ∈XM. 2.5

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Define

eXQX − r

nX, X∈XM, EX, Y ge X, Y , X, Y∈XM.

2.6

Then, consider

ES− r

n g. 2.7

The0,2tensorEis called tensor of Einstein14with respect to the metric connection∇. If Sis symmetric, thenEis also symmetric.

Definition 2.1. A metric connection∇ with symmetric Ricci tensorSwill be called a “Ricci- symmetric metric connection.”

Example 2.2. In a semi-Riemannian manifoldM, g, a semisymmetric metric connection∇of Yano8is given by

∇XY ∇XYuYX−gX, YU, X, Y ∈XM, 2.8

where∇is Levi-Civita connection,Uis a vector field, anduis its associated 1 form given by uX gX, U. The Ricci tensorSwith respect to∇is given by

SS−n−2α−traceαg, 2.9

whereSis the Ricci tensor, andαis a0,2tensor field defined by

αX, Y ∇XuY−uXuY 1

2uUgX, Y, X, Y ∈XM. 2.10

The Ricci tensorSis symmetric if 1 form,uis closed.

Example 2.3. Anε-almost para contact metric manifoldM, ϕ, ξ, η, g, εis given by ϕ² I−η⊗ξ, ηξ 1, g

ϕX, ϕY

gX, Y−εηXηY, 2.11

whereϕis a tensor field of type1,1,ηis 1 form,ξis a vector field andε±1. Anε-almost para contact metric manifold satisfying

∇Xϕ

Y −g

ϕX, ϕY

ξ−εηYϕ²X 2.12

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is called anε-para Sasakian manifold15. In anε-para Sasakian manifold, the semisymmetric metric connection∇given by

∇XY ∇XYηYX−gX, Yξ 2.13

is a Ricci symmetric metric connection.

Example 2.4. An almost contact metric manifoldM, ϕ, ξ, η, gis given by ϕ² −Iη⊗ξ, ηξ 1, g

ϕX, ϕY

gX, Y−ηXηY, 2.14

whereϕis a tensor field of type1,1,ηis 1-form andξis a vector field. An almost contact metric manifold is a Kenmotsu manifold16if

∇Xϕ Y g

ϕX, Y

ξ−ηYϕX, 2.15

and is a Sasakian manifold17if ∇Xϕ

Y gX, Yξ−ηYX. 2.16

In an almost contact metric manifoldM, the semisymmetric metric connection∇given by

∇XY ∇XYηYX−gX, Yξ 2.17

is a Ricci symmetric metric connection ifMis Kenmotsu, but the connection fails to be Ricci symmetric ifMis Sasakian.

LetM, gbe ann-dimensional semi-Riemannian manifold equipped with a metric connection∇. A symmetric 0,2tensor fieldH, which is covariantly constant with respect to∇, is called a special quadratic first integral for brevity SQFI 18with respect to∇. The semi-Riemannian metricgis always an SQFI. A set of SQFI tensors{H1, . . . , H}with respect to∇ is said to be linearly independent if

c₁H₁· · ·cH0, c₁, . . . , c ∈R 2.18 implies that

c₁· · ·c 0. 2.19

The set{H1, . . . , H}is said to be a complete set if any SQFI tensorHwith respect to∇ can be written as

Hc₁H₁· · ·cH, c₁, . . . , c ∈R. 2.20

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The “index” 4 of the manifoldMwith respect to ∇, denoted by i_∇, is defined to be the numberof members in a complete set{H1, . . . , H}.

We will need the following Lemma.

Lemma 2.5. Let M, g be an n-dimensional semi-Riemannian manifold equipped with a Ricci symmetric metric connection∇. Then the following statements are true.

aIf∇XS0, then∇XE0. Conversely, ifris constant and∇XE0 then∇XS0.

bIf ∇XS /0 and ψ is a nonvanishing diﬀerentiable function such that ψ∇XSand g are linearly dependent, then∇XE0.

The proof is similar to Lemmas 1.2 and 1.3 in7for a semisymmetric metric connection and is therefore omitted.

3. Quasiconformal Curvature Tensor

LetM, gbe ann-dimensionaln > 3semi-Riemannian manifold equipped with a metric connection∇. The conformal curvature tensor Cwith respect to the∇ is defined by19, page 90as follow:

CX, Y, Z, V RX, Y, Z, V − 1 n−2

SY, ZgX, V −SX, ZgY, V

gY, ZSX, V −gX, ZSY, V r

n−1n−2

gY, ZgX, V−gX, ZgY, V ,

3.1

and the concircular curvature tensorZwith respect to∇ is defined by20,21, page 87as follows:

ZX, Y, Z, V RX, Y, Z, V − r nn−1

gY, ZgX, V−gX, ZgY, V

. 3.2

As a generalization of the notion of conformal curvature tensor and concircular curvature tensor, the quasiconformal curvature tensorC∗with respect to∇ is defined by9as follows:

C_∗X, Y, Z, V aRX, Y, Z, V b

SY, Zg X, V−SX, ZgY, V gY, ZSX, V −gX, ZSY, V

− r n

a

n−12b gY, ZgX, V−gX, ZgY, V ,

3.3

whereaandbare constants. In fact, we have

C∗X, Y, Z, V −n−2bCX, Y, Z, V a n−2bZX, Y, Z, V . 3.4

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Since there is no restrictions for manifolds ifa0 andb0, therefore it is essential for us to consider the case ofa /0 orb /0. From3.4it is clear that ifa1 andb−1/n−2, then C∗C; if a1 andb0, thenC∗Z.

Now, we need the following.

Definition 3.1. A semi-Riemannian manifoldM, gequipped with a metric connection∇ is said to be

a∇-quasiconformally flat if C_∗0, b∇-conformally flat if C0, and c∇-concircularly flat if Z0.

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 manifoldM, gequipped with a metric connection∇ is said to be

a∇-quasiconformally symmetric if ∇ C∗0, b∇-conformally symmetric if ∇C0, and c∇-concircularly symmetric if ∇Z0.

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 M be a semi-Riemannian manifold of dimension n > 2. Then M is ∇- quasiconformally flat if and only if one of the following statements is true:

ia n−2b0,a /0/b, andMis∇-conformally flat,

iia n−2b /0,a /0,Mis∇-conformally flat, and ∇-concircularly flat, iiia n−2b /0,a0 and Ricci tensorSwith respect to∇ satisfies

S− r

n g 0, 3.5

whereris the scalar curvature with respect to∇.

Proof. UsingC_∗0 in3.3, we get 0aRX, Y, Z, V b

SY, ZgX, V −SX, ZgY, V gY, ZSX, V −gX, ZSY, V

− r n

a

n−1 2b

gY, ZgX, V−gX, ZgY, V ,

3.6

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from which we obtain the following:

a n−2b S− r

n g

0. 3.7

Case 1a n−2b 0 anda /0/b. Then from3.3and3.1, it follows thatn− 2bC0, which givesC0. This gives the statementi.

Case 2a n−2b /0 anda /0. Then from3.7 SY, Z r

ngY, Z. 3.8

Using3.8in3.6, we get

a

RX, Y, Z, V − r nn−1

0. 3.9

Sincea /0, then by3.2Z 0 and by using3.9,3.8in3.1, we getC0. This gives the statementii.

Case 3 n−2b /0 anda0, we get3.5. This gives the statementiii. Converse is true in all cases.

Corollary 3.4see22, Theorem 5.1. LetMbe a semi-Riemannian manifold of dimensionn >2.

ThenMis quasiconformally flat if and only if one of the following statements is true:

ia n−2b0,a /0/b, andMis conformally flat, iia n−2b /0,a /0,andMis of constant curvature, and iiia n−2b /0,a0, andMis Einstein manifold.

Remark 3.5. In23, 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 con- stanta /0.

c 23, Corollary 1.2. A quasiconformally flat manifold is Einstein if the constants a0 andb /0.

However, the converses need not be true in these three results. But, inCorollary 3.4we get a complete classification of quasiconformally flat manifolds.

4. ∇-Quasiconformally Symmetric Manifolds

Let M, g be an n-dimensional semi-Riemannian manifold equipped with the metric connection∇. Let R be the curvature tensor ofMwith respect to the metric connection∇.

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IfH is a parallel symmetric0,2tensor with respect to the metric connection ∇, then we easily obtain that

H

∇UR

X, YZ, V H

Z,

∇UR

X, YV

0, X, Y, Z, V, U∈XM. 4.1

The solutionsH of4.1is closely related to the index of quasiconformally symmetric and concircularly symmetric manifold with respect to the∇.

Lemma 4.1. Let M, g be an n-dimensional semi-Riemannian ∇-quasiconformally symmetric manifold,n >2 andb /0. Then

trace

∇UE

0. 4.2

Proof. Using2.7in3.3, we get the following:

C_∗X, Y, Z, V aRX, Y, Z, V b

EY, Zg X, V−EX, ZgY, V gY, ZEX, V −gX, ZEY, V

− ar nn−1

gY, ZgX, V−gX, ZgY, V .

4.3

Taking covariant derivative of4.3and using∇UC∗0, we get

a

∇UR

X, Y, Z, V b

∇UE

X, ZgY, V−

∇UE

Y, ZgX, V

−gY, Z

∇UE

X, V gX, Z

∇UE Y, V

a

∇Ur nn−1

gY, ZgX, V−gX, ZgY, V .

4.4

Contracting4.4with respect toY andZand using2.2, we get

a

∇US

X, V − btrace

∇UE

gX, V

− n−2b

∇UE

X, V a

∇Ur

n gX, V.

4.5

Using4.5, we get4.2.

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Theorem 4.2. IfM, gis ann-dimensional semi-Riemannian∇-quasiconformally symmetric man- ifold,n >2 andb /0, then4.1takes the form

det

⎛

⎝HX, Z−1

ntraceHgX, Z HY, V− 1

ntraceHgY, V ∇UE

X, Z

∇UE Y, V

⎞

⎠0. 4.6

If∇UE /0, then4.6has the general solution HUX, Y f

∇US

X, Y 1 n

traceHU−f

∇Ur

gX, Y, 4.7

wherefis an arbitrary nonvanishing diﬀerentiable function.

Proof. Using4.4in4.1, we get 0b

∇UE

X, ZHY, V−

∇UE

Y, ZHX, V

− gY, ZH

∇Ue X, V

gX, ZH

∇Ue Y, V

∇UE

X, VHY, Z−

∇UE

Y, VHX, Z

−gY, VH

∇Ue X, Z

gX, VH

∇Ue Y, Z

a

∇Ur nn−1

gY, ZHX, V−gX, ZHY, V

gY, VHX, Z−gX, VHY, Z .

4.8

Let{e1, . . . , e_n}be an orthonormal basis of vector fields inM. TakingX Ze_iin4.8and summing up tonterms, then, using4.2, we have

0b

n−1H

∇Ue Y, V

H

∇Ue V, Y

−traceH

∇UE

Y, V−gY, Vⁿ

i1

H

∇Ue ei, ei

a

∇Ur nn−1

traceHgY, V−nHY, V .

4.9

Interchanging Y and V in 4.9 and subtracting the so-obtained formula from 4.9, we deduce that

H

∇Ue Y, V

H

∇Ue V, Y

. 4.10

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Now, interchangingXandZ,Y, andV in4.8and taking the sum of the resulting equation and4.8and using4.9and4.10, we get4.6. If∇UE /0, then using2.7leads to4.7.

Theorem 4.3. IfM, gis ann-dimensional semi-Riemannian∇-quasiconformally symmetric man- ifold,n >2 andb /0, and if there is a vector fieldUso that

∇UE0 , ∇Ur /0, 4.11

then the solution of 4.1isHf g, wherefis a diﬀerentiable nonvanishing function.

Proof. Using4.11,4.8becomes

gY, ZHX, V−gX, ZHY, V gY, VHX, Z−gX, VHY, Z 0, 4.12 InterchangingXandZ,Y andV in4.12and taking the sum of the resulting equation and 4.12, we get

gX, ZHY, V−gY, VHX, Z 0. 4.13

Therefore, the tensor fieldsHandgare proportional.

Theorem 4.4. Let M, g be an n-dimensional semi-Riemannian ∇-quasiconformally symmetric manifold,n >2 andb /0. If there is a vector fieldUsatisfying the condition4.11, theni_∇ 1.

Proof. ByTheorem 4.3and from the fact that∇Ug 0 and ∇UH 0, it follows thatf is constant. Thus,i_∇ 1.

Theorem 4.5. Let M, g be an n-dimensional semi-Riemannian ∇-quasiconformally symmetric manifold,n > 2 andb /0, for which the tensor fieldE is not covariantly constant with respect to the Ricci symmetric metric connection∇. If i_∇ >1, then there is a vector fieldU, so that the equation

∇UH0 4.14

has the fundamental solutions

H1g, H2 ψ∇US, 4.15

whereψis a diﬀerentiable nonvanishing function.

Proof. Given that∇UE /0, there isUso that the tensorial equation4.1has general solution which depends onU.gis obviously a solution of4.14because∇Ug 0,galso satisfies the tensorial equation4.1, andH_U given by4.7is also a solution of4.14. Equation4.14 has at least two solution asi_∇ > 1. These two solutions are independent. ByLemma 2.5b ψ∇USandg are independent and we get two fundamental solution of ∇UH 0 which is H₁g, H₂ψ∇US, where ψis a diﬀerentiable nonvanishing function.

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Theorem 4.6. Let M, g be an n-dimensional semi-Riemannian ∇-quasiconformally symmetric manifold,n > 2 andb /0, for which the tensor fieldE is not covariantly constant with respect to the metric connection∇. Then 1 ≤i_∇ ≤n1.

Proof. LetU_i,i1, . . . , pbe independent vector fields, for which

∇UiE /0, 4.16

and letψ_i∇UiSandgbe the fundamental solutions of∇UiH0. Obviouslyp < n, asU_iare independent. Therefore, we havep1 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. IfM, gbe ann-dimensional semi-Riemannian∇-concircularly symmetric manifold, then the4.1takes the form

det

HX, Z HY, V gX, Z gY, V

0. 4.17

Proof. Taking covariant derivative of3.2and using∇UZ0, we get

∇UR

X, Y, Z, V ∇Ur nn−1

, 4.18

which, when used in4.1, yields

0 ∇Ur nn−1

gY, ZHX, V−gX, ZHY, V

gY, VHX, Z−gX, VHY, Z .

4.19

Now, we interchangeX withZ and Y withV in4.19 and take the sum of the resulting equation and4.19, and we get4.17.

Theorem 4.9. LetM, gbe ann-dimensional semi-Riemannian∇-concircularly symmetric mani- fold. Theni_∇ 1.

Proof. ByTheorem 4.8and from the fact that∇Ug0 and∇UH0, we geti_∇ 1.

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5. Discussion

A semi-Riemannian manifold is said to be decomposable 4 or locally reducible if there always exists a local coordinate systemxⁱso that its metric takes the form

ds² ^r

a,b1

g_abdx^adx^b ⁿ

α,βr1

g_αβdx^αdx^β, 5.1

whereg_abare functions ofx¹, . . . , x^r andg_αβare functions ofx^r1, . . . , xⁿ. 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 irreducible4. 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 ofsemi-Riemannian manifolds. For example, a Riemannian manifold is decomposable if and only if its index is greater than one 4.

Moreover, a complete Riemannian manifold is reducible if and only if its index is greater than one4. A second-order0,2-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 metricg if and only if the manifold is reducible1, that is the index of the manifold is greater than 1. In 1951, Patterson24found a similar result for semi-Riemannian manifolds. In fact, he proved that a semi-Riemannian manifoldM, gadmitting a special Killing tensor K_ij, other than g, is reducible if the matrix Kij 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.

ByTheorem 4.6, we conclude that a∇-quasiconformally symmetric Riemannian man- ifoldwhere∇ is any Ricci symmetric metric connection, not necessarily Levi-Civita connec- tionis 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 manifoldMⁿ, gis1/12nn1²n2, which is attained if and only ifMis 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 curvature25. But, from Theorem 4.6, we also conclude that the maximum number of linearly independent special Killing tensors in a 4-dimensional Robertson-Walker spacetime11, page 341is 5.

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