On Contact Metric R-Harmonic Manifolds

K.Arslan, C. Murathan, C. Özgür and A. Yildiz

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

Abstract

In this paper we consider contact metric R-harmonic manifolds M with ξ belonging to (κ, μ)-nullity distribution. In this context we have κ ≤ 1. If κ < 1, then M is either locally isometric to the product E^{n+1} × S^{n}(4), or locally isometric to E(2) (the group of the rigid motions of the Euclidean 2-space). If κ = 1, then M is an Einstein-Sasakian manifold.

Mathematics Subject Classification: 53C05, 53C20, 53C21, 53C25. Key words: contact metric manifold, Einstein manifold, (k,μ)-nullity distribution, R- harmonic manifold

1 Introduction

Throughout this paper we use the notations and terminology of [1] and [2]. Let M be a (2n + 1)–dimensional Riemannian C^{∞} manifold. M^{2n+1} is said to be contact manifold, if it admits a global differential 1-form η such that η∧(dη)^{n} ≠ 0, everywhere on M^{2n+1}. Given a contact form η, we have a unique vector field ξ, which is called the characteristic vector field, satisfying η(ξ) = 1, dη(ξ, X) = 0, for any vector field X.

It is well–known that, there exists a Riemannian metric g and a (1,1)–tensor field φ such that

η(X) = g(X, ξ), dη(X, Y) = g(X, φY) and φ^{2}X = −X + η(X)ξ, (1)

where X and Y are vector fields on M^{2n+1}.

From (1) it follows that η ∘ φ = 0, φ(ξ) = 0, g(φX, φY) = g(X, Y) − η(X)η(Y). A Riemannian manifold M^{2n+1} equipped with structure tensors (φ, ξ, η, g) satisfy-

ing (2) is said to be a contact metric manifold and denoted by M = (M^{2n+1}, φ, ξ, η, g). Given a contact metric manifold M we can define a (1, 1)-tensor field h by h =

\frac{1}{2}L_{ξ}φ, where L denotes Lie differentiation. Then we may observe that h is symmetric and satisfies hξ = 0 and hφ = −φh, ∇_{X}ξ = −φX − φhX, where ∇ is Levi-Civita connection [2]. A contact metric manifold for which ξ is Killing vector field is called K -contact manifold. It is well-known that a contact manifold is K-contact if and only if h = 0.

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 1-6 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

We denote by R the Riemannian curvature tensor field defined by

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

for all vector fields X, Y, Z.

For a contact metric manifold M one may define naturally an almost complex structure on M × R . If this almost complex structure is integrable, M is said to be a Sasakian manifold [1]. A Sasakian manifold is characterized by the condition (∇_{X}φ)Y = g(X, Y)ξ − η(X)Y, for all vector fields X and Y on the manifold [1].

Let M be a contact metric manifold. It is well known that M is Sasakian if and only if R(X, Y)ξ = η(Y)X − η(X)Y, (2)

for all vector fields X and Y [1].

A contact metric manifold M is said to be η− Einstein if

Q = aId + bη ⊗ ξ, (3)

where Q is the Ricci operator and a, b are smooth functions on M [2].

2 Contact metric manifolds with ξ belonging to (κ, μ)–nullity distribution

In this section we give some well-known results.

Let M be a contact metric manifold. The (κ, μ)- nullity distribution of M for the pair (κ, μ) is a distribution

\begin{aligned}N(κ, μ) : p → N_{p}(κ, μ) & = & \{Z ∈ T_{p}M | R(X, Y)Z = \\ & = & κ[g(Y, Z)X − g(X, Z)Y]+ \\ & + & μ[g(Y, Z)hX − g(X, Z)hY]\}, \end{aligned} (4)

where κ, μ ∈ R (see [5]). So if the characteristic vector field ξ belongs to the (κ, μ)- nullity distribution we have

R(X, Y)ξ = κ [η(Y)X − η(X)Y] + μ [η(Y)hX − η(X)hY] .

Lemma 2.1 [2]. If M is a contact metric manifold with ξ belonging to the (κ, μ)- nullity distribution, then

(∇_{X}h)Y = [(1 − κ)g(X, φY) − g(X, hφY)] ξ + η(Y)h(φX + φhX) − μη(X)φhY,

where X and Y are any vector fields on M. Theorem 2.2 [2]. Let M be a contact metric manifold with ξ belonging to a (κ, μ)- nullity distribution. Then κ ≤ 1. If κ = 1, then h = 0 and M is Sasakian manifold. If κ < 1, M admits three mutually orthogonal and integrable distributions D(0), D(λ) and D(-λ) determined by the eigenspaces of h, where λ = \sqrt{1 − κ}.

Lemma 2.3 [2]. Let M be a contact metric manifold with ξ belonging to the (κ, μ)- nullity distribution (κ < 1). For any vector field X, the Ricci operator Q is given by

\begin{matrix}QX = [2(n − 1) − nμ]X + [2(n − 1) + μ]hX+ \\ +[2(1 − n) + n(2κ + μ)]η(X)ξ; n ≥ 1. \end{matrix} (5)

A consequence of Lemma 2.3 is the following Lemma 2.4. Let M be a contact metric manifold. If ξ belonging to the (κ, μ)-nullity distribution, then

\begin{matrix}(∇_{X}S)(Y, Z) = [2(n − 1) + μ]g((∇_{X}h)Y, Z)+ \\ + [2(1 − n) + n(2κ + μ)] \{g(Y, ∇_{X}ξ)η(Z) + g(Z, ∇_{X}ξ)η(Y)\} . \end{matrix} (6)

3 R–Harmonic manifolds

Let M be a (2n+1)–dimensional Riemannian C^{∞} manifold, ∇ and R denote its Levi– Civita derivative and curvature tensor respectively.

A tensor field R of type (1,3) 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

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

Proposition 3.1 [4]. Let R be an algebraic curvature tensor field which satisfies the second Bianchi identity. If S is the associated Ricci tensor field, then

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

Definition 3.1. An algebraic curvature tensor field R is harmonic (or Codazzi type in the sense of [3]) if

(divR)(X, Y, Z) = 0.

A Riemannian manifold M is called R-harmonic if its curvature tensor field R is harmonic.

It is obvious that every Ricci-symmetric manifold (i.e. ∇S = 0) is R-harmonic. Corollary 3.2 [4]. An algebraic curvature tensor field satisfying the second Bianchi identity is harmonic if and only if the associated Ricci tensor Q (related to S by S(X, Y) = g(QX, Y)) is a Codazzi tensor field i.e., (∇_{X}Q)Y − (∇_{Y} Q)X = 0, for every X, Y ∈ χ(M).

Now we state our main results. Theorem 3.3. Let M be a contact metric R-harmonic manifold with ξ belonging to (κ, μ)-nullity distribution.

i) If κ < 1, then M is either a) locally isometric to the product E^{n+1} × S^{n}(4), or b) locally isometric to E(2) (the group of the rigid motions of the Euclidean 2-space).

ii) If κ = 1, then M is an Einstein-Sasakian manifold. Proof. i) Since M is a contact metric manifold with ξ belonging to (κ, μ)-nullity distribution, with κ < 1, then by the covariant differentiation of the relation (13) we have

\begin{matrix}(∇_{X}Q)Y − (∇_{Y} Q)X = [2(n − 1) + μ] [(∇_{X}h)Y − (∇_{Y} h)X] + \\ + [2(1 − n) + n(2κ + μ)] [g(Y, ∇_{X}ξ)ξ + η(Y)∇_{X}ξ] + \\ − [2(1 − n) + n(2κ + μ)] [g(X, ∇_{Y} ξ)ξ + η(X)∇_{Y} ξ] . \end{matrix} (7)

By Lemma 3.1 iv) in [2] it can be seen that

\begin{aligned}(∇_{X}h)Y − (∇_{Y} h)X & = & (1 − k) [2g(X, φY)ξ + η(X)φY − η(Y)φX] \\ & + & (1 − μ) [η(X)φhY − η(Y)φhX] . \end{aligned} (8)

Substituting (8) into (7) and using R-harmonic property we obtain

\begin{matrix}0 = (∇_{X}Q)Y − (∇_{Y} Q)X = \\ = [2(n − 1) + μ] \{(1 − k) [2g(X, φY)ξ + η(X)φY − η(Y)φX] \\ +(1 − μ) [η(X)φhY − η(Y)φhX]\}+ \\ + [2(1 − n) + n(2κ + μ)] [g(Y, ∇_{X}ξ)ξ+ \\ +η(Y)∇_{X}ξ − g(X, ∇_{Y} ξ)ξ − η(X)∇_{Y} ξ \end{matrix} (9)

Taking the product of both sides of the equation (9) by ξ and using the fact that φ is antisymmetric, h is symmetric, φξ = 0, ∇_{X}ξ = −φX −φhX, after some computation we find [κ(2 − μ) + μ(n + 1)] g(X, φY) = 0. Since g(X, φY) = dη(X, Y) ≠ 0, we have κ(2 − μ) + μ(n + 1) = 0.

Taking X = ξ into (9) and using the fact that φ is antisymmetric, h is symmetric, φξ = 0, ∇_{X}ξ = −φX − φhX, after some computations we obtain

[κ(2 − μ) + μ(n + 1)] φY + [2nκ + μ(3 − n − μ)] φhY = 0. (10)

Since κ(2−μ)+μ(n+1) = 0, the relation (10) becomes [2nκ + μ(3 − n − μ)] φhY =0. So we have two possible cases:

Case I. κ(2 − μ) + μ(n + 1) = 0 and [2nκ + μ(3 − n − μ)] = 0.

Case II. φhY = 0.

Let us consider these in turn.

(Case I). Suppose κ(2 − μ) + μ(n + 1) = 0 and [2nκ + μ(3 − n − μ)] = 0. Then solving this system we obtain the following solutions:

κ = μ = 0, κ = μ = 3 + n or κ = \frac{(n − 1)(n + 1)}{n}, μ = 2 − 2n.

For the case κ = μ = 0, M must be locally isometric to the product E^{n+1} × S^{n}(4) (see [1] p.121). Since κ < 1, the case κ = μ = 3 + n is not possible. But the case

κ = \frac{(n − 1)(n + 1)}{n}, μ = 2−2n is possible only for n = 1. Thus M is 3-dimensional in this case and by Theorem 3 in [2], M is locally isometric to E(2) (the rigid motions of the Euclidean 2-space).

(Case II). Suppose φhY = 0. Then we have ∇_{Y} ξ = −φY which implies that M is K−contact. Therefore h = 0. Since h^{2} = (κ − 1)φ^{2}, we obtain k = 1 which is contradicting the fact that κ < 1 so this case does not occur.

ii) If κ = 1, then M is an Einstein-Sasakian manifold.

First, using the relation (∇_{X}S)(Y, Z) = ∇_{X}S(Y, Z) − S(∇_{X}Y, Z) − S(Y, ∇_{X}Z) and the symmetric property of Q one can write g(Y, (∇_{X}Q)Z) = g((∇_{X}Q)Y, Z) and similarly g(X, (∇_{Y} Q)Z) = g((∇_{Y} Q)X, Z). Since M is R-harmonic, by Corollary 3.2 we obtain g((∇_{X}Q)Y, Z)) = g((∇_{Y} Q)X, Z). (11)

Setting Z = ξ into (11) and using the relations ∇_{X}ξ = −φX and Qξ = 2nξ (see [1]), we have

− 2ng(Y, φX) + g(Y, QφX) = −2ng(X, φY) + g(X, QφY). (12)

Since M is Sasakian, we have Qφ = φQ. So the equation (12) becomes

2ng(X, φY) − g(X, QφY) = 0. (13)

Interchanging Y with φY in (13) one finds S(X, Y) = 2ng(X, Y), i.e., M is an Einstein manifold. This completes the proof of the theorem. Corollary 3.4. Let M be a contact metric manifold with ξ belonging to (κ, μ)-nullity distribution . If M is R-harmonic on the distribution D = {X | η(X) = 0, X ∈ χ(M)}, then M is either Einstein or Einstein-Sasakian manifold. Proof. Suppose M is a contact metric manifold with ξ belonging to (κ, μ)-nullity distribution.

First we suppose that κ < 1. If M is a R-harmonic on the distribution D, then the equations (5) and (7) respectively become

QX = [2(n − 1) − nμ] X + [2(n − 1) + μ] hX, (14)

(∇_{X}Q)Y − (∇_{Y} Q)X = [2(n − 1) + μ] [(∇_{X}h)Y − (∇_{Y} h)X] = 0.

So we have the following cases.

Case I. 2(n − 1) + μ = 0,

Case II. (∇_{X}h)Y − (∇_{Y} h)X = 0.

Let us consider these in turn.

(Case I). Suppose 2(n − 1) + μ = 0. Then the equation (14) becomes QX = [2(n − 1) − nμ] X, which implies that M is an Einstein manifold.

(Case II). Suppose (∇_{X}h)Y − (∇_{Y} h)X = 0. Then by Lemma 2.1 we have (∇_{X}h)Y − (∇_{Y} h)X = 2(1 − κ)g(X, φY) = 0, which implies κ = 1. This contradicts the fact that κ < 1. So this case does not occur.

If κ = 1, then by the same discussion given in Theorem 3.3 ii) it is easy to show that M is an Einstein manifold. This completes the proof of the corollary.

References

[1] D. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Math. Springer-Verlag, Berlin-Heidelberg-New-York, 509 (1976).

[2] D. Blair, T. Koufogiorgos and B.J. Papantoniou, Contact Metric Manifolds Satisfying a Nullity Condition, Israel J. Math. 91(1995),189-214.

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

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

[5] B.J. Papantoniou, Contact Metric Manifolds Satisfying R(ξ, X).R = 0 and ξ ∈ (k, μ)− nullity distribution, Yokohama Math.J. 40(1993), 149-161.

K. Arslan, C. Murathan and C. Özgür Uludag University Faculty of Art and Sciences Görükle/Bursa/TURKEY e-mail: arslan@@uludag.edu.tr

A. Yildiz Dumlupinar University Faculty of Art and Sciences Kutahya/TURKEY