On Pseudo Ricci-Symmetric Manifolds

K. Arslan, R. Ezentaş, C. Murathan and C. Özgür

Abstract In the present study we consider pseudo Ricci-symmetric manifolds in the sense of M. C. Chaki. We show that pseudo Ricci-symmetric manifolds satisfying divR = 0 (resp.divC = 0) property are Einstein (resp.Ricci flat) manifolds.

Mathematics Subject Classification: 53C20, 53C25 Key words: R-harmonic manifold, Pseudo Ricci-symmetric manifold

1 Introduction

Let M be an n-dimensional (n ≥ 3) Riemannian manifold. For the vector fields X, Y, Z ∈ χ(M) and the the Levi-Civita connection ∇ of M the curvature tensor R and the Ricci operator S of M are defined by

R(X, Y)Z = ∇_{X}(∇_{Y} Z) − ∇_{Y} (∇_{X}Z) − ∇_{[X,Y]}Z,

and

S(X, Y) = g(SX, Y)

respectively. Furthermore for the vector field W the Riemannian Christoffel curvature tensor R of M is defined by R(X, Y, Z, W) = g(R(X, Y)Z, W) [4].

Let Π be a non-degenerate tangent plane to M at p ∈ M given by X, Y ∈ χ(M). Then the sectional curvature K(Π) of Π defined by

K(X, Y){g(X, X)g(Y, Y) − g(X, Y)^{2}} = g(R(X, Y)Y, X)

which is independent of the choice of the basis X, Y for Π.

A tensor field R of type (1, 2) on M is called algebraic curvature tensor field if it has symmetric properties of the curvature tensor field of Riemannian manifolds.

The curvature tensor R satisfies the second Bianchi identity if

(1.1)  (∇_{X}R)(Y, Z, W) + (∇_{Y} R)(X, Z, W) + (∇_{Z}R)(X, Y, W) = 0.

Let R be an algebraic curvature tensor field which satisfies the second Bianchi identity. If S is the associated Ricci tensor field then

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 1-5 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1.2)  (divR)(X, Y, Z) = (∇_{X}S)(Y, Z) − (∇_{Y} S)(X, Z).

For a Riemannian manifold M if the Ricci tensor S is of the form S = λg then it is called Einstein space [4]. If S = 0 then M is called Ricci-flat.

The Weyl conformal curvature tensor C is defined by

C(X, Y, Z, W) = R(X, Y, Z, W) − \frac{1}{n − 2}\{g(X, W)S(Y, Z) + g(Y, Z)S(X, W)−
− g(X, Z)S(Y, W) − g(Y, W)S(X, Z)}+
+ \frac{τ}{(n − 1)(n − 2)}\{g(X, W)g(Y, Z) − g(X, Z)g(Y, W)\}.

An algebraic curvature tensor field R is harmonic (or Codazzi type in the sense of [7]) if (divR)(X, Y, Z) = 0. A Riemannian manifold M is called R-harmonic if its curvature tensor field R is harmonic [1]

The divergence divC of the Weyl conformal curvature tensor C is defined by

(1.3)  \begin{aligned}(divC)(X, Y, Z) & = & \frac{n − 3}{n − 2}\{(∇_{X}S)(Y, Z) − (∇_{Y} S)(X, Z)\}− \\ & − & \frac{1}{2(n − 1)}\{g(Y, Z)∇_{X}τ − g(X, Z)∇_{Y} τ\}. \end{aligned}

In the present study we consider pseudo Ricci-symmetric submanifolds and also hypersurfaces.

The notion of pseudo Ricci-symmetric (PRS) manifolds were introduced by M. C. Chaki in 1987. A non-flat Riemannian manifold (M^{n}, g) (n > 2) is called pseudo Ricci-symmetric if its Ricci tensor S is not identically zero and satisfies

(1.4)  (∇_{X}S)(Y, Z) = 2α(X)S(Y, Z) + α(Y)S(X, Z) + α(Z)S(Y, X),

where α is a 1-form which is non-zero for every X, Y, Z ∈ χ(M) and ∇ being operator of covariant differentiation with respect to the metric g [2]. In [3] the authors considered conformally flat pseudo Ricci-symmetric manifolds, see also [5] for the case M is a contact manifold.

In the present study we consider pseudo Ricci-symmetric manifolds. We show that pseudo Ricci-symmetric manifolds satisfying divR = 0 (resp.divC = 0) property are Einstein (resp.Ricci flat) manifolds.

2 Pseudo Ricci-Symmetric manifolds

Let (M, g), (n ≥ 3), be an n-dimensional Riemannian manifold and e_{i}, e_{j} (1 ≤ i, j ≤ n) orthonormal vector fields tangent to M and K_{ij} is the sectional curvature of a plane spanned by the vectors e_{i} and e_{j}. Then by definition of S we have

(2.1)  S(e_{i}, e_{i}) = ∑_{k=1}^{n} g(R(e_{k}, e_{i})e_{i}, e_{k}) = ∑_{k=1}^{n} K_{ik},

(2.2)  S(e_{j}, e_{j}) = ∑_{k=1}^{n} K_{jk}, S(e_{i}, e_{j}) = 0.

First we prove the following result. Proposition. Let M^{n} be a n-dimensional Riemannian manifold. If M is pseudo Riccisymmetric then

(2.3)  ∑_{k=1}^{n}(K_{ik} − K_{jk})g(e_{j}, ∇_{e_{i}}e_{i}) = α(e_{j}) ∑_{k=1}^{n} K_{ik},

(2.4)  e_{i} [∑_{k=1}^{n} K_{ik}] = 4α(e_{i}) ∑_{k=1}^{n} K_{ik},

(2.5)  e_{i} [∑_{k=1}^{n} K_{jk}] = 2α(e_{i}) ∑_{k=1}^{n} K_{jk}.

Proof. Let e_{i}, e_{j} be orthonormal vector fields tangent to M. Combining (2.1)-(2.2) and (1.4) we find

(2.6)  (∇_{e_{i}}S)(e_{i}, e_{j}) = α(e_{j})S(e_{i}, e_{i}),

(2.7)  (∇_{e_{i}}S)(e_{i}, e_{i}) = 4α(e_{i})S(e_{i}, e_{i}),

(2.8)  (∇_{e_{i}}S)(e_{j}, e_{j}) = 2α(e_{i})S(e_{j}, e_{j}).

Moreover, from the covariant differentiation of S we have

(2.9)  (∇_{e_{i}}S)(e_{i}, e_{j}) = −S(∇_{e_{i}}e_{i}, e_{j}) − S(e_{i}, ∇_{e_{i}}e_{j}),

(2.10)  (∇_{e_{i}}S)(e_{i}, e_{i}) = ∇_{e_{i}}S(e_{i}, e_{i}),

(2.11)  (∇_{e_{i}}S)(e_{j}, e_{j}) = ∇_{e_{i}}S(e_{j}, e_{j}) − 2S(∇_{e_{i}}e_{j}, e_{j}).

By the use of (2.1)-(2.2) we get

(2.12)  S(∇_{e_{i}}e_{i}, e_{j}) = ∑_{k=1}^{n} g(R(e_{k}, ∇_{e_{i}}e_{i})e_{j}, e_{k}) = ∑_{k=1}^{n} K_{jk}g(e_{j}, ∇_{e_{i}}e_{i}),

(2.13)  S(e_{i}, ∇_{e_{i}}e_{j}) = ∑_{k=1}^{n} K_{jk}g(e_{j}, ∇_{e_{i}}e_{i}), S(∇_{e_{i}}e_{i}, e_{i}) = 0.

Combining (2.13), (2.13) and (2.9) we obtain

(2.14)  (∇_{e_{i}}S)(e_{i}, e_{j}) = (∑_{k=1}^{n}(K_{ik} − K_{jk})) g(e_{j}, ∇_{e_{i}}e_{i}).

Forthermore differentiating (2.1) and (2.2) covariantly we have

(2.15)  (∇_{e_{i}}S)(e_{i}, e_{i}) = e_{i} [∑_{k=1}^{n} K_{ik}] ,

(2.16)  (∇_{e_{i}}S)(e_{j}, e_{j}) = e_{i} [∑_{k=1}^{n} K_{jk}] (or (∇_{e_{j}}S)(e_{i}, e_{i}) = e_{j} [∑_{k=1}^{n} K_{ik}]).

Since the left hand sides of the equations (2.6)-(2.8) are equal to the left hand sides of (2.14)-(2.16) we get the result. □ Theorem 2.2. Let M be a n-dimensional pseudo Ricci-symmetric manifold. If M is R -harmonic (i.e divR = 0) then it is Ricci-flat. Proof. Let M is R-harmonic so divR = 0. Using (1.2) we have

(2.17)  (∇_{e_{i}}S)(e_{i}, e_{j}) − (∇_{e_{j}}S)(e_{i}, e_{i}) = 0.

Making use of (2.7), (2.8) and (2.1) the equation (2.17) reduces to α(e_{j})S(e_{i}, e_{i}) = 0. Since α is a non-zero one form then S(e_{i}, e_{i}) = 0. Thus M is Ricci flat this complates the proof. □ Theorem. Let M be a n-dimensional pseudo Ricci-symmetric manifold. If divC = 0 then M^{n} is an Einstein manifold. Proof. Suppose divC = 0. Then by (1.3) we have

(2.18)  (∇_{e_{i}}S)(e_{i}, e_{j}) − (∇_{e_{j}}S)(e_{i}, e_{i}) + \frac{1}{2} \frac{n − 2}{(n − 1)(n − 3)}e_{j}[τ] = 0.

where ∇_{e_{j}}τ = e_{j}[τ]. Substituting (2.14) and (2.16) into (2.18) we obtain

(2.19)  ∑_{k=1}^{n}(K_{ik} − K_{jk})g(e_{j}, ∇_{e_{i}}e_{i}) − e_{j}[∑_{k=1}^{n} K_{ik}] + \frac{1}{2} \frac{n − 2}{(n − 1)(n − 3)}e_{j}[τ] = 0.

Moreover, substituting (2.3) and (2.5) into (2.19) we get

−α(e_{j}) ∑_{k=1}^{n} K_{ik} + \frac{1}{2} \frac{n − 2}{(n − 1)(n − 3)}e_{j}[τ] = 0.

By the use of (2.1) the above equation becomes

(2.20)  α(e_{j})S(e_{i}, e_{i}) = \frac{1}{2} \frac{n − 2}{(n − 1)(n − 3)}e_{j}[τ].

On the other hand

(2.21)  ∑_{i=1}^{n} α(e_{j})S(e_{i}, e_{i}) = \frac{1}{2} \frac{n − 2}{(n − 1)(n − 3)}e_{j}[τ] ∑_{i=1}^{n} g(e_{i}, e_{i}),

which implies

(2.22)  e_{j}[τ] = \frac{2(n − 1)(n − 3)}{n(n − 2)}α(e_{j})τ.

However combining (2.22) and (2.21) one can get S(e_{i}, e_{i}) = \frac{1}{n}τ. Thus M is an Einstein manifold. □ Theorem 2.4. Let M be a n-dimensional pseudo Ricci-symmetric manifold whose one of family of curvature lines consists of geodesic (i.e. ∇_{e_{i}}e_{i} = 0). Then M^{n} is Ricci flat.

Proof. Putting ∇_{e_{i}}e_{i} = 0 into (2.3) we get α(e_{i}) ∑_{k=1}^{n} K_{ik} = 0. Since the one form α is

non-zero then by (2.1) one can get S(e_{i}, e_{i}) = 0, which means that M is Ricci flat. □ Acknowledgements. This study is supported by Uludağ University Research Fund.

References

[1] K. Arslan, C. Murathan, C. Özgür, and A. Yildiz, On contact metric R-harmonic manifolds, B.J.G.A. 5 (2000), 1-6.

[2] M. C. Chaki, On Contact metric R-harmonic manifolds, Analele Stiintifice Ale Universitatii AL. I. Cuza, Iasi, 33 (1987), 53-58.

[3] M. C. Chaki and P. Chakrabarti, On conformally flat pseudo Ricci symmetric manifolds, Tensor, N. S., 52(1993), 217-222.

[4] B.Y. Chen, Geometry of Submanifolds, Dekker 1973.

[5] U. C. De, J. C. Ghosh and B. Barua, On pseudo symmetric and pseudo Ricci symmetric contact manifolds, Bull. Cal. Math. Soc. 89 (1997), 305-310.

[6] Memerthzheim, M. and Recziegel, H, Hypersurface with Harmonic Curvature in Space of Constant Curvature, Cologne, March 1993.

[7] Mukhopadhyay, S. and Barua, B, On a Type of Non-flat Riemannian Manifold, Tensor, 56 (1995), 227-232.

Kadri Arslan, Rıdvan Ezentaş and Cihan Özgür

Cengizhan Murathan

Department of Mathematics  Department of Mathematics
Uludağ University  Balıkesir University
16059 Bursa, TURKEY  10100, Balıkesir, TURKEY
E-mail: arslan@uludag.edu.tr  E-mail: cozgur@balikesir.edu.tr