Curvature Tensors on A-Einstein Sasakian Manifolds

G.P. Pokhariyal

Abstract The author and Mishra [1] have introduced some curvature tensors to study their physical and geometric properties. In this paper, W_{2}-curvature tensor, its associated symmetric and skew-symmetric tensors are studied in an A-Einstein Sasakian manifold.

Mathematics Subject Classification: 53C25, 53B21, 53C15. Key words: W_{2}-curvature tensor, A-Einstein Sasakian manifold, symmetric and skew symmetric tensors.

1 Introduction

Let us consider an n-dimensional real differential manifold M_{n}. Let there exist a vector valued linear function F, a 1-form A and a vector field T, satisfying

(1.1)  (a)  \overline{\overline{X}} + X = A(X)T, (b)  \overline{X} \underline{\underline{def}} F (X)

for any arbitrary vector field X. Then M_{n} is called an almost contact manifold, and the structure (F, T, A) is called an almost contact structure.

From (1.1) we have, rank(F) = n − 1, n is odd, say 2m + 1, and that \overline{T} = 0,

(1.2)  A(\overline{X}) = 0,  A(T) = 1.

In addition, if in M_{n} there exists a metric tensor g satisfying

(1.3a)  g(\overline{X}, \overline{Y}) = g(X, Y) − A(X)A(Y),

then M_{n} is called an almost Grayan manifold. From (1.1) and (1.3)a, we have

(1.3b)  g(X, T) = A(X).

Putting ′F (X, Y) = g(\overline{X}, Y) then, we have

(1.4)  \begin{matrix}(a) & ′F (\overline{X}, \overline{Y}) = −g(X, \overline{Y}) = g(\overline{X}, Y) = ′F (X, Y) \\ (b) & ′F (X, Y) +′ F (Y, X) = 0 \end{matrix}

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 45-50 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

If in an almost Grayan Manifold,

′F (X, Y) = (D_{X}A)(Y) − (D_{Y} A)(X) = (dA)(X, Y),

where D is a Riemannian connexion, then M_{n} is called an almost Sasakian manifold. In a Sasakian manifold, F is closed. An almost Sasakian manifold is said to be Sasakian manifold, if T is killing vector:

(D_{X}A)(Y) + (D_{Y} A)(X) = 0.

Thus in a Sasakian manifold

′F (X, Y) = (D_{X}A)(Y) and (D_{X} ′F)(Y, Z) = ′R(X, Y, Z, T),

where R is the curvature tensor of the type (0, 4) of M_{n}. In a Sasakian manifold, we have [2]

(1.5)  \begin{matrix}(a) & ′R(T, X, Y, T) = g(\overline{X}, \overline{Y}) = g(X, Y) − A(X)A(Y) \\ (b) & ′R(X, Y, Z, T) = A[R(X, Y, Z)] = A(X)g(Y, Z) − A(Y)g(X, Z) \\ (c) & ′R(T, X, Y, Z) = A(Z)g(X, Y) − g(X, Z)A(Y) \\ (d) & ′R(T, Y, Z) = g(Y, Z)T − A(Z)Y \\ (e) & ′R(X, Y, T) = A(Y)X − A(X)Y, \end{matrix}

where ′R(X, Y, Z, U) = g[R(X, Y, Z), U] and

(1.6)  \begin{matrix}(a) & Ric(X, T) = g(r(X), T) = A(r(X)) = (n − 1)A(X) \\ (b) & Ric(\overline{X}, Y) + Ric(X, \overline{Y}) = 0, \end{matrix}

where Ric is the Ricci tensor.

The Sasakian manifold M_{n} is called an A-Einstein Sasakian manifold, if the Ricci tensor satisfies [3]

(1.7)  Ric(X, Y) = αg(X, Y) + βA(X)A(Y)

for some scalar fields α and β.

The author and Mishra [1] have defined a tensor (1.8)

′W_{2}(X, Y, Z, U) = ′R(X, Y, Z, U) + \frac{1}{n − 1}[g(X, Z)Ric(Y, U) − g(Y, Z)Ric(X, U)]

This tensor is skew symmetric in X and Y, therefore, breaking it into skew symmetric part and symmetric part in Z and U , we obtained

(1.9)  \begin{matrix}′E(X, Y, Z, U) = R(X, Y, Z, U) + \frac{1}{2(n − 1)}[g(X, Z)Ric(Y, U)− \\ −g(Y, Z)Ric(X, U) − g(X, U)Ric(Y, Z) + g(Y, U)Ric(X, Z)] \end{matrix}

(1.10)  \begin{matrix}′F (X, Y, Z, U) = \frac{1}{2(n − 1)}[g(X, Z)Ric(Y, U)− \\ −g(Y, Z)Ric(X, U) + g(X, U)Ric(Y, Z) − g(Y, U)Ric(X, Z)] \end{matrix}

The tensor ′E(X, Y, Z, U) possesses all the symmetric and skew-symmetric properties of ′R(X, Y, Z, U) as well as the cyclic property. The tensor ′F (X, Y, Z, U) satisfies only the cyclic property with fixed U.

2 Properties of tensors

In this section we study the properties of W_{2}, E and F curvature tensors in A-Einstein Sasakian manifolds. Theorem 2.1. In an A-Einstein Sasakian manifold, we have (2.1)

\begin{matrix}(a) & ′W_{2}(T, Y, Z, U) = \frac{A(Z)}{n − 1}[(α − 1)g(Y, U) + βA(Y)A(U)], \\ (b) & ′W_{2}(T, \overline{Y} , Z, \overline{U}) = [\frac{α − 1}{n − 1}][g(Y, U) − A(Y)A(U)]A(Z), \\ (c) & ′W_{2}(\overline{X}, \overline{Y} , \overline{Z}, \overline{U}) −′ W_{2}(X, Y, Z, U) = (\frac{α}{n − 1} − 1)A(Z)A(Y)g(X, U). \end{matrix}

Proof. From (1.8) we have

′W_{2}(T, Y, Z, U) =′ R(T, Y, Z, U) + \frac{1}{n − 1}[g(T, Z)Ric(Y, U) − g(Y, Z)Ric(T, U)].

Using (1.3)b,(1.5)c,(1.6)a, and (1.7), we get (2.1)a. Barring Y and U in (2.1)a and using (1.2), we get (2.1)b. Barring all the vector fields in (1.8) and using (1.2),(1.3),(1.5) and (1.7), we get (2.1)c. □ Corollary 2.1. For A-Einstein Sasakian manifold, we have

′W_{2}(T, Y, \overline{Z}, U) = 0,

′W_{2}(T, \overline{Y} , Z, U) = [\frac{α − 1}{n − 1}]A(Z)g(\overline{Y} , U),

′W_{2}(T, Y, Z, \overline{U}) = [\frac{α − 1}{n − 1}]A(Z)g(Y, \overline{U}),

′W_{2}(T, \overline{Y} , Z, U) +′ W_{2}(T, Y, Z, \overline{U}) = 0.

Proof. Using (1.2), (1.3)b, (1.4)a, (1.7) and (1.8), we get the results. □ Theorem 2.2. In an A-Einstein Sasakian manifold, we have

(2.2)  \begin{matrix}(a) & ′E(T, Y, Z, U) = \frac{1}{2}[1 − \frac{α}{n − 1}]\{A(U)g(Y, Z) − A(Z)g(Y, U)\}, \\ (b) & ′E(X, Y, Z, T) = \frac{1}{2}[1 − \frac{α}{n − 1}]\{A(X)g(Y, Z) − A(Y)g(X, Z)\}, \\ (c) & ′E(T, Y, Z, T) = \frac{1}{2}[g(Y, Z)(1 − \frac{α}{n − 1}) − \frac{β}{n − 1}A(Y)A(Z)]. \end{matrix}

Proof. Using (1.9), (1.3)b, (1.5), (1.6) and (1.7), we get the results. □ Corollary 2.2. For the A-Sasakian manifold, we have

′E(T, Y, Z, \overline{U}) = −\frac{1}{2}[1 − \frac{α}{n − 1}]A(Z)g(Y, \overline{U}),

′E(\overline{Y} , Z, U, T) = −\frac{1}{2}[1 − \frac{α}{n − 1}]A(Z)g(\overline{Y} , U),

′E(\overline{Y} , Z, U, T) + ′E(T, Y, Z, \overline{U}) = 0,

′E(T, \overline{Y} , \overline{Z}, T) = \frac{1}{2}[1 − \frac{α}{n − 1}]\{g(Y, Z) − A(Y)A(Z)\}.

Proof. Using (1.2), (1.3)b, (1.4)a, (1.7) and (1.9), we get the results. □ Theorem 2.3. For an A-Einstein Sasakian manifold, we have

(2.3)  \begin{aligned}(a) & ′F (X, Y, Z, T) = \frac{β}{2(n − 1)}[g(X, Z)A(Y) − g(Y, Z)A(X)] \\ (b) & ′F (T, Y, Z, U) = \frac{β}{2(n − 1)}[2A(Y)A(Z)A(U)− \\ & −A(U)g(Y, Z) − A(Z)g(Y, U)] \\ (c) & ′F (T, Y, Z, T) = \frac{β}{2(n − 1)}[A(Y)A(Z) − g(Y, Z)]. \end{aligned}

Proof. Using (1.10), (1.3)b, (1.6) and (1.7), we get the result. □ Corollary 2.3. In an A-Einstein Sasakian manifold, we have

′F (\overline{X}, Y, Z, T) = \frac{β}{2(n − 1)}[g(\overline{X}, Z)A(Y)]

′F (\overline{X}, \overline{Y} , Z, T) =′ F (T, Y, \overline{Z}, \overline{W}) = 0

′F (T, \overline{Y} , \overline{Z}, T) = \frac{−β}{2(n − 1)}[g(Y, Z) − A(Y)A(Z)]

′F (T, Y, Z, \overline{U}) = \frac{−β}{2(n − 1)}[g(Y, \overline{U}) − A(Z)]

′F (\overline{Y} , Z, U, T) +′ F (T, Y, Z, \overline{U}) = 0.

Proof. Using (1.10), (1.2), (1.3) and (1.7), we get the results. □

3 Symmetric A-Einstein Sasakian manifolds

We consider an A-Einstein Sasakian manifold M_{n} and define the following Definition 3.1. The manifold M_{n} is called E-Symmetric and F-symmetric provided

(3.1)  (a) (D_{Y} E)(Z, U, V) = 0, and (b) (D_{Y} F)(Z, U, V) = 0

are satisfied, where D_{Y} denotes the covariant differentiation. From (3.1), we have

R(X, Y, E(Z, U, V))−E((R(X, Y, Z), U, V)−E(Z, R(X, Y, U), V)−E(Z, U, R(X, Y, V)) = 0.

This equation implies

\begin{matrix}′R(T, Y, E(Z, U, V), T) −′ E((R(T, Y, Z), U, V, T)− \\ −′E(Z, R(T, Y, U), V, T) − ′E(Z, U, R(T, Y, V), T) = 0. \end{matrix}

Using (1.3), (1.5), (1.6) and (1.9), we get

\begin{matrix}′E(Z, U, V, Y) + \frac{1}{2}[g(Y, U)\{g(Z, V) − \frac{Ric(Z, V)}{n − 1}\} − g(Y, Z)\{g(U, V) − \frac{Ric(U, V)}{n − 1}\}+ \\ +A(Z)A(V)\{g(Y, U) − \frac{Ric(Y, U)}{n − 1}\} − A(U)A(V)\{g(Y, Z) − \frac{Ric(Y, Z)}{n − 1}\}] = 0. \end{matrix}

Using (1.7), we find

′E(Z, U, V, Y) + \frac{1}{2}(1 − \frac{α}{n − 1})\{g(Y, U)g(Z, V)−

−g(Y, Z)g(U, V) + A(Z)A(V)g(Y, U) − A(U)A(V)g(Y, Z)}+

+\frac{β}{2(n − 1)}\{g(Y, Z)A(U)A(V) − g(Y, U)A(Z)A(V)\} = 0.

Thus, we have Theorem 3.1. If the A-Einstein Sasakian manifold M_{n} is E-Symmetric, then E is given by

(3.2)  \begin{matrix}′E(Z, U, V, Y) = \frac{1}{2}(1 − \frac{α}{n − 1})\{g(Y, Z)g(U, V)− \\ −g(Y, U)g(Z, V) − A(Z)A(V)g(Y, U) + A(U)A(V)g(Y, Z)\}+ \\ +\frac{β}{2(n − 1)}[g(Y, U)A(Z)A(V) − g(Y, Z)A(U)A(V)\}. \end{matrix}

On similar lines, we have the following theorem. Theorem 3.2. If an A-Einstein Sasakian manifold M_{n} is F-Symmetric, then F is given by

(3.3)  ′F (Z, U, V, Y) = \frac{β}{2(n − 1)}[g(Y, U)g(Z, V) − g(Y, Z)g(U, V)].

Proof. The proof follows the pattern of Theorem 3.1. □

4 Discussion

The W_{2}-curvature tensor was introduced on the line of Weyl curvature tensor and by breaking W_{2} into skew-symmetric parts the tensor E has been defined. Rainich conditions for the existence of the non-null electrovariance can be obtained by W_{2} and E, if we replace the matter tensor by the contracted part of these tensors. The tensor E enables to extend Pirani formulation of gravitational waves to Einstein space [4]. In Sasakian manifold and other manifolds, tensors W_{2} and E satisfy properties, some of which are similar to that of Weyl’s projective tensor and conformal curvature tensor respectively. Thus, these tensors can alternatively be used to study physical and geometrical properties of manifolds. Acknowledgement. I am thankful to research student B.M. Nzimbi and L.K. Nyongesa for typing the paper.

References

[1] G.P.Pokhariyal and R.S. Mishra, Curvature tensors and their relativities significance, Yokohama Math.Jour, 18 (1970), 105-108.

[2] G.P. Pokhariyal, Study of a new curvature tensor in a Sasakian manifold, Tensor, NS, 36 (1982), 222-226.

[3] M.Yamata, On an η-Einstein Sasakian manifold satisfying the certain conditions, Tensor, N.S, 49 (1990), 305-309.

[4] G.P. Pokhariyal, Relative Significance of curvature tensors, Internet. J. Math & Math. Sci., Vol 5, No.1 (1982), 133-139.

Department of Mathematics

University of Nairobi

P.O. Box 30197, Nairobi, KENYA e-mail: pokhariyal@iconnect.co.ke