On Conformally Flat Pseudosymmetric Spaces

U. C. De and S.K. Ghosh

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

In a recent paper [1] M. C. Chaki introduced and studied a type of nonflat Riemannian space (M^{n}, g)(n ≥ 2) whose curvature tensor R_{ijk}^{h} satisfies the condition

(1)  R_{ijk,l}^{h} = 2λ_{l}R_{ijk}^{h} + λ^{h}R_{lijk} + λ_{i}R_{ilk}^{h} + λ_{k}R_{ijl}^{h}

where λ_{i} is a non-zero vector and comma denotes covarient differentiation with respect to the metric g_{ij}. Such a space was called by him a pseuso symmetric space, the vector λ_{i} was called its associated vector and an n-dimensional space of this kind has been denoted (P S)_{n}. Tarafder[2] proved that a conformally flat (P S)_{n} (n ≥ 3) with non-zero constant scalar curvature is a subprojective space in the sense of Kagan[3], if the associated vector is gradient. In the present paper we obtain the above result without assuming any restriction on the scalar curvature. Among others it is shown that a conformally flat (P S)_{n} can be expressed as a warped product I × e^{q}M^{∗} where M^{∗} is an Einstein space and such space is a space of quasi-constant curvature [4].

Mathematics Subject Classification: 53C25 Key words: conformally flat space, subprojective space, Einstein space

1 Conformally flat (P S)_{n} (n ≥ 3)

It is known [1] that a conformally flat (P S)_{n} (n ≥ 3) can not be of zero scalar curvature and also it is known [2] that in a conformally flat (P S)_{n}

R_{ij} = \frac{R − t}{n − 1}g_{ij} + \frac{nt − R}{(n − 1)λ_{p}λ^{p}} λ_{i}λ_{j}

where R denotes the scalar curvature and t is a scalar.

The above expression can be written as

(1.1)  R_{ij} = αg_{ij} + βv_{i}v_{j}

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where α = \frac{R − t}{n − 1}, β = \frac{nt − R}{n − 1} are two scalars and v_{i} = \frac{λ_{i}}{\sqrt{λ_{i}λ^{i}}} is a unit vector. On

the otherhand, a conformally flat space is conformally symmetric, that is, C_{ijk,l}^{h} = 0. The above equation is equivalent to

(1.2)  R_{jl,k} − R_{jk,l} = \frac{1}{2(n − 1)}(g_{jl}R_{,k} − g_{jk}R_{,l}).

The relation (1.1) implies

(1.3) R_{ij,k} = α_{k}g_{ij} + β_{k}v_{i}v_{j} + β(v_{j}v_{i,k} + v_{i}v_{j,k}), where α_{,k} = α_{k} and β, k = β_{k}.

Substituting (1.3) into (1.2) we obtain

(1.4) \begin{aligned}α_{k}g_{jl} & + & β_{k}v_{j}v_{l} + β(v_{l}v_{j,k} + v_{j}v_{l,k}) − α_{l}g_{jk} − β_{l}v_{j}v_{k} − β(v_{k}v_{j,l} + v_{j}v_{k,l}) \\ & = & \frac{1}{2(n − 1)}(g_{jl}R_{k} − g_{jk}R_{l}) \end{aligned}

where R_{k} = R_{,k}.

Since v^{i}v_{i} = 1 and (v_{i,k})v^{i} = 0, so by transvecting with g^{jl}, (1.4) reduces to

(1.5)  (n − 1)α_{k} + β_{k} − (β_{a}v^{a})v_{k} − β(v_{k}v_{,a}^{a} + v^{a}v_{k,a}) = \frac{1}{2} R_{k}.

Transvecting (1.4) with v^{j} we obtain

(1.6) (α_{k}v_{l} − α_{l}v_{k}) + (β_{k}v_{l} − β_{l}v_{k}) + β(v_{l,k} − v_{k,l}) = \frac{1}{2(n − 1)}(v_{l}R_{k} − v_{k}R_{l}).

Transvecting again with v^{l} we have

(α_{k} + β_{k}) − (α_{a}v^{a})v_{k} − (β_{a}v^{a})v_{k} − βv^{a}v_{k,a} = \frac{1}{2(n − 1)} \{R_{k} − (v^{a}R_{a})v_{k}\}

Substituting this into (1.5) we find

(1.7)  (n − 2)α_{k} − βv_{k}v_{,a}^{a} + (α_{a}v^{a})v_{k} − \frac{1}{2(n − 1)}v_{k}(v^{a}R_{a}) = \frac{1}{2} \frac{n − 2}{(n − 1)}R_{k}

Transvecting (1.7) with v^{k}, we get

(n − 1)(α_{a}v^{a}) − βv_{,a}^{a} = \frac{1}{2}(R_{a}v^{a}).

Thus (1.7) reduces to

(1.8)  R_{k} = λv_{k} + 2(n − 1)(α_{k} − μv_{k})

where λ = R_{a}v^{a} and μ = α_{a}v^{a}.

Substituting this into (1.6), we obtain

(1.9)  (β_{k}v_{l} − β_{l}v_{k}) + β(v_{l,k} − v_{k,l}) = 0.

Now if v_{i} is gradient, that is, v_{l,k} − v_{k,l} = 0, then

(1.10)  β_{k}v_{l} − β_{l}v_{k} = 0. T hat is,

(1.10a)  β_{k} = av_{k} where a is a scalar.

Now by (1.8), (1.9) and (1.10) the equation (1.4) reduces to

β(v_{l}v_{j,k} − v_{k}v_{j,l}) = \frac{1}{2(n − 1)}ϕ(v_{k}g_{jl} − v_{l}g_{jk})

where ϕ = λ − 2(n − 1)μ.

Transvecting the above equation with v^{l} and using v_{j,l} = v_{l,j}, we get

(1.11)  v_{j,k} = \frac{1}{2(n − 1)} \frac{ϕ}{β} (v_{k}v_{j} − g_{jk}).

Let us consider the scalar function

f = \frac{1}{2(n − 1)} \frac{ϕ}{β} ≠ 0.

We have

f_{k} = −\frac{1}{2(n − 1)} \frac{ϕ}{β^{2}} β_{k} + \frac{1}{2(n − 1)} \frac{ϕ_{k}}{β} where f_{k} = f_{,k} and ϕ_{k} = ϕ_{,k}.

Again (1.8) implies

R_{k,j} = ϕ_{j}v_{k} + ϕv_{k,j} + 2(n − 1)(α_{k,j} − α_{j,k})

from which we get ϕ_{j}v_{k} = ϕ_{k}v_{j,} that is, ϕ_{k} = Av_{k,} where A is a scalar function. Thus form (1.10a) and (1.12) f_{k} = Bv_{k} where

B = \frac{1}{2(n − 1)β} (−\frac{ϕ_{a}}{β} + A)

Using (1.12), it is easy to show that ω_{i} = \frac{1}{2(n − 1)β} ϕv_{i} is a gradiant vector field.

In fact, ω_{i,j} = v_{i}f_{j} +fv_{ij} = βv_{j}v_{i} +fv_{ij} = ω_{j,i}. Thus (1.11) can be written as follows: v_{j,k} = −fg_{jk} + ω_{k}v_{j} where ω_{k} is gradient.

Hence v_{i} is a concircular vector field. Since f ≠ 0, v_{i} is a proper concircular vector. Hence λ_{i} is a proper concircular vector field. It is known [3] that if a conformally flat space admits a proper concircular vector

field, then the space is a subprojective space in the sence of Kagan. Thus we can state Theorem 1. If the associated vector of a conformally flat (P S)_{n} is gradient, then the space is a subprojective space.

In [6] K. Yano proved that in order that a Riemannian space admits a concircular vector field, it is necessary and sufficient that there exists a coordinate system with respect to which the fundamental quadratic differential form may be written in the form

ds^{2} = (dx^{1})^{2} + c^{q}g_{αβ}^{∗}dx^{α}dx^{β}

where g_{αβ}^{∗} = g_{αβ}^{∗}(x^{ν}) are the functions of x^{γ} only (α, β, γ = 2, 3..........n) and q = q(x^{1}) ≠ constant is a function of x^{1} only. Since conformally flat (P S)_{n} admits proper concircular vector field v_{i} the space under consideration is the warped product 1 × e^{q}M^{∗} where (M^{∗}, g^{∗}) is an (n − 1)-dimensional Riemannian space. Gebarowski [6] proved that the warped product 1 × e^{q}M^{∗} satisfies (1.2) iff M^{∗} is an Einstein space. Thus we state the following theorem : Theorem 2. A conformally flat (P S)_{n} is the warped product 1 × e^{q}M^{∗} where M^{∗} is an Einstein space.

A conformally flat Riemannian space is said to be of quasi-constant curvature [4] if the curvature tensor R_{hijk} is given by

(1.11) R_{hijk} = a(g_{hj}g_{ik} − g_{hk}g_{ij}) + b(g_{hj}θ_{i}θ_{k} − g_{hk}θ_{i}θ_{j} − g_{ij}θ_{h}θ_{k} + g_{ik}θ_{h}θ_{j})

where a and b are differentiable functions and θ_{i} is a unit vector. Since our space is conformally flat, the curvature tensor is given by

R_{hijk} = \frac{1}{n − 2} (R_{hk}g_{ij} − R_{hj}g_{ik} + R_{ij}g_{hk} − R_{ik}g_{hj}) − \frac{R}{(n − 1)(n − 2)}(g_{ij}g_{hk} − g_{ik}g_{hj})

Now on account of (1.1) the above equation reduces to (1.11), where

θ_{i} = v_{i,} a = \frac{R}{(n + 1)(n − 2)} − \frac{2α}{(n − 2)} and b = −\frac{β}{n − 2}

Hence we obtain Theorem 3. A conformally flat (P S)_{n} is a space of quasi-constant curvature.

References

[1] M.C. Chaki, On Pseudosymmetric Manifolds, Analele Stiintifice Ale Universitatii “Al. I. Cuza” Iasi, 33 (1987), 53-58.

[2] M. Tarafder, On Conformally flat Pseudosymmetric Manifolds, Analele Stiintifice Ale Universitatii “Al. I. Cuza” Iasi, 1995, 237-241.

[3] T. Aditi, On Subprojective space - III, Tohoku Math J. 3 (1951), 343-358.

[4] B.Y. Chen, and K. Yano, Hypersurface of a Conformally Flat Space, Tensor, N. S., 26 (1972), 318-322.

[5] K. Yano, On the Torseforming direction in Riemannian Spaces, Proc. Imp. Acad. Tokyo, 20 (1944), 340-345.

[6] A. Gebarowski, Nearly Conformally Symetric Wraped Product Manifolds, Bulletin of the Institute of Mathematics, Academia Sinica, 20, No. 4 (1992), 359-371.

Department of Mathematics

University of Kalyani

Kalyani - 741235, W. B., India

Email:ucde@klyuniv.ernet.in