ON DECOMPOSABLE QUASI-EINSTEIN SPACES ABSOS ALI SHAIKH and SHYAMAL KUMAR HUI The object of the present paper is to classify decomposable quasi-Einstein spaces with an example.

ABSOS ALI SHAIKH and SHYAMAL KUMAR HUI

The object of the present paper is to classify decomposable quasi-Einstein spaces with an example.

AMS 2000 Subject Classification: 53B30, 53B50, 53C15, 53C25.

Key words: quasi-Einstein space, decomposable quasi-Einstein space, scalar cur- vature, Killing vector.

1. INTRODUCTION

It is well known that a Riemannian space Vⁿ, n > 2, is Einstein if its Ricci tensor R_ij of type (0, 2) is of the form

Rij =p gij, where pis a constant, which turns into

Rij = R ngij,

R being the scalar curvature (constant) of the space.

The notion of quasi-Einstein spaces arose during the study of exact solu- tions of the Einstein field equations as well as during considerations of quasi- umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein spaces. A Riemannian space Vⁿ, n > 2, is said to be quasi- Einstein space ([1], [3], [4], [5], [6], [7], [10], [11]) if its Ricci tensor R_ij of type (0,2) is not identically zero and satisfies the condition

(1.1) Rij =pgij+qAiAj,

where p, qare scalars of which,q 6= 0 andA_i is a nowhere vanishing covariant vector defined byAi =ρ^jgij,ρⁱ being a unit vector called the generator of the space. An n-dimensional space of this kind is denoted by (QE)_n. The scalars p, q are known as the associated scalars.

The object of the present paper is to classify the decomposable quasi- Einstein spaces. The paper is organized as follows. Section 2 is concerned with preliminaries. Section 3 deals with decomposable (QE)n and a full classifica- tion of such spaces is given. It is proved that in a decomposable (QE)_n, one

MATH. REPORTS13(63),1 (2011), 89–94

among the decompositions is Einstein and the other one is quasi-Einstein. Es- sentially, this implies that in the decomposable case, the generator vector field is tangent to one of the factors. And the result is also true when one of the two factors is one dimensional.

A Riemannian spaceVⁿis said to be Ricci parallel if its Ricci tensorR_ij of type (0, 2) satisfies the condition R_ij,k = 0, where denotes the covariant differentiation with respect to the metric tensor g. The class of Ricci parallel spaces is very natural generalization of the class of spaces of constant scalar curvature. Again, by the decomposition of the covariant derivative R_ij,k of the Ricci tensor R_ij of type (0, 2), Gray [9] introduced two important classes A, B, which lie between the class of Ricci-parallel spaces and the spaces of constant scalar curvature, namely (i) the class Ais the class of spaces whose Ricci tensor is cyclic parallel and (ii) the class B is the class of spaces whose Ricci tensor is of Codazzi type [8]. The spaces of class Aare said to be cyclic Ricci parallel spaces. In a (QE)n with cyclic parallel Ricci tensor, the scalar curvature is always constant but the converse is not true, in general. That is, in a (QE)_n with constant scalar curvature, the associated scalars p and q are not necessarily constants. However, if p and q are constants then such a space is of constant scalar curvature. It is proved that if the Ricci tensor of a non-Einstein decomposition of a decomposable (QE)n is cyclic parallel, then its associated scalars are constants and the generator ρⁱ is a Killing vector.

Finally an example of a non-Einstein decomposable (QE)n is given.

2. DECOMPOSABLE RIEMANNIAN SPACES

In this section, some formulas are derived, which will be useful to the study of decomposable (QE)_n.

A Riemannian space Vⁿ is said to be decomposable space [12] if it can be expressed as V₁^r×V₂^n−r for 2 ≤ r ≤ n−2, that is, in some coordinate neighbourhood of the Riemannian space Vⁿ, the metric can be expressed as (2.1) ds²=g_ijdxⁱdx^j = ˜g_abdx^adx^b+g^∗_αβ dx^αdx^β,

where ˜gab are functions of x¹, x², . . . , x^r denoted by ˜x and g^∗_αβ are functions of x^r+1, x^r+2, . . . , xⁿ denoted by x;^∗ a, b, c, . . . run from 1 tor and α, β, γ, . . . run from r+ 1 ton. The two parts of (2.1) are the metrics ofV₁^r,r≥2, and of V₂^n−r,n−r≥2, which are called the decompositions of the decomposable space Vⁿ=V₁^r×V₂^n−r, 2≤r ≤n−2.

LetVⁿ be a decomposable Riemannian space such thatVⁿ=V₁^r×V₂^n−r for 2≤r ≤n−2. Here throughout the paper each object denoted by a ‘tilde’

is assumed to be from V₁^r and each object denoted by a ‘star’ is assumed to be from V₂^n−r.

Then ([12]) from (2.1) we have

g_ab= ˜g_ab, g_αβ =g^∗_αβ, g^ab = ˜g^ab, g^αβ =g^∗αβ, gaα= 0 =g^aα, Γ^c_ab = ˜Γ^c_ab, Γ^γ_αβ =

∗

Γ^γ_αβ, R_αbcd= 0 =R_aβcδ=R_aβγδ, Rabcd,α= 0 =Raβcδ,k =Raβcδ,µ, Rabcd = ˜Rabcd, Rαβγδ=R^∗αβγδ,

Rab= ˜Rab, Rαβ =R^∗αβ, Rab,c = ˜Rab,c, Rαβ,γ =R^∗αβ,γ, and

R=g^ijRij = ˜g^abR˜_ab+g^∗αβR^∗αβ= ˜R+R,^∗

where R, ˜R, and R^∗ are respectively scalar curvatures ofV,V₁,V₂.

3. DECOMPOSABLE(QE)n

Let us consider a Riemannian space Vⁿ which is decomposable (QE)_n. Then Vⁿ=V₁^r×V₂^n−r, 2≤r ≤n−2. Now, from (1.1) we have

R˜_ab=p g˜_ab+q A_aA_b, (3.1)

∗

Rαβ =p g^∗_αβ +q A_αA_β (3.2)

and

q A_αA_b = 0, which implies that

(3.3) AαA_b = 0.

From (3.3) it follows that either A_α= 0 orA_b = 0.

IfAα = 0, then from (3.2) we get

∗

Rαβ=pg^∗_αβ,

which shows that the decomposition V2 is an Einstein space.

IfA_b = 0, then from (3.1) we get R˜_ab =p ˜g_ab,

which shows that the decomposition V₁ turns into Einstein space. Thus we can state

Theorem 3.1.A decomposable Riemannian space is(QE)_nif and only if one among the decompositions is Einstein and the other one is quasi-Einstein.

The Einstein part of a decomposable (QE)nis always satisfies cyclic pa- rallel Ricci tensor. We now consider the other non-Einstein part of a decom- posable (QE)n, with cyclic parallel Ricci tensor. Without loss of generality, we may assume that Vⁿ=V₁^r×V₂^n−r, 2≤r≤n−2, is (QE)_nof whichV₁ is non-Einstein. Then we have

(3.4) R˜_bc,a+ ˜R_ca,b+ ˜R_ab,c = 0.

From (3.1) it follows that

(3.5) R˜ =rp+q

and

(3.6) R˜_bc,a=p_,a˜g_bc+q_,aA_bA_c+q

A_b,aA_c+A_bA_c,a . By (3.6), (3.4) yields

p,a˜gbc+q,aAbAc+p,bg˜ca+q,bAcAa+p,c˜gab+q,cAaAb+ (3.7)

+q

A_b,aA_c+A_bA_c,a+A_c,bA_a+A_cA_a,b+A_a,cA_b+A_aA_b,c

= 0.

Transvecting (3.7) with ρ^bρ^c, we obtain (3.8) 2qA_a,mρ^m=−

p_,a+q_,a+ 2A_a{(p_,m+q_,m)}ρ^m . Again, contracting (3.7) over band c, we get

(3.9) (r+ 2)p,a+q,a+ 2Aaq,m ρ^m+ 2q

Aa,m ρ^m+Aa r

X

b=1

A_b,b

= 0.

Now, from (3.6), we have

(3.10) R˜_,a=rp_,a+q_,a.

In a (QE)n with cyclic parallel Ricci tensor, the scalar curvature is always constant. Hence

(3.11) R˜,a = 0.

By (3.11) we have from (3.10) that

(3.12) p_,a =−1

r q_,a. Using (3.12) in (3.8), we have

(3.13) 2qAa,m ρ^m =−r−1 r

q,a+ 2Aa q,mρ^m . By (3.12) and (3.13), (3.9) yields

(3.14) 2q Aa

r

X

b=1

A_b,b= r+ 1 r q,a−2

rAaq,m ρ^m.

Again, transvecting (3.14) with ρ^a, we obtain

(3.15) 2q

r

X

b=1

A_b,b = r−1

r q_,m ρ^m. Also transvecting (3.13) with ρ^a, we get

(3.16) q_,mρ^m= 0.

By (3.16), it follows from (3.13) and (3.15) that

(3.17) 2qAa,mρ^m=−r−1

r q,a

and (3.18)

r

X

b=1

A_b,b = 0.

Using (3.12) and (3.16)–(3.18) in (3.9), we get

(3.19) q_,a= 0 for alla

and

(3.20) p,a = 0 for all a,

that is, p andq are constant. This leads to

Theorem 3.2. If the Ricci tensor of the non-Einstein decomposition of a decomposable(QE)_nis cyclic parallel, then its associated scalars are constants.

Next, by (3.19) and (3.20), (3.7) implies

(3.21) A_b,aAc+A_bAc,a+A_c,bAa+AcA_a,b+Aa,cA_b+AaA_b,c = 0.

Also, by (3.19) it follows from (3.13) that Aa,mρ^m= 0.

Transvecting (3.21) with ρ^c we obtain by virtue of above that

(3.22) A_b,a+A_a,b= 0,

which implies that ρ^b is a Killing vector. This leads to

Theorem 3.3.If the Ricci tensor of the non-Einstein decomposition of a decomposable(QE)nis cyclic parallel, then the generatorρ^b is a Killing vector.

Example. If V₁ is an Einstein space and V₂ is a canal hypersurface in Rⁿ, that is, the envelope of a 1-parameter family of hypersurfaces [2], then the Riemannian product V =V1×V2 is a decomposable quasi-Einstein space which is not a decomposable Einstein space.

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Received 10 May 2009 University of Burdwan

Department of Mathematics Burdwan – 713104 West Bengal, India [email protected]