On φ-Conformally Flat Contact Metric Manifolds

K. Arslan, C. Murathan and C. Özgür

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

Abstract This paper presents a study of contact metric manifolds, namely φ -conformally flat ones under the condition that the charecteristic vector field ξ belongs to the (k, μ)-nullity distribution.

Mathematics Subject Classification: 53C05, 53C20, 53C21, 53C25 Key words: Contact metric manifold, φ-conformally flat manifold, η-Einstein Manifold, Sasakian Manifold

1 Introduction

The conformally symmetric K-contact manifolds was considered in [7]. It has shown that conformally symmetric K-contact manifold is locally isometric to the unit sphere. However in [8] the ξ-conformally flat contact metric manifolds are considered, and it has shown that such manifolds are φ-Einstein-Sasakian. Although, in [3] it has shown that a compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S^{1}-bundle over an almost Kaehler space of constant halomorphic sectional curvature (see Theorem 3.2 in [3]).

In this paper we shall therefore consider φ-conformally flat contact metric manifold under the condition that the characteristic vector field ξ belongs to the (k, μ)-nullity distribution (i.e. (k, μ)-contact manifold). We generalize the (Theorem 3.2) of [3].

This paper is organized as follows; In section 2 we give some preliminaries. In section 3 we give some known results. Finally in section 4 we prove the following theorem. Theorem 4.3. Let M be a contact metric manifold under the condition that ξ belongs to (k, μ)-nullity condition. If M is φ-conformally flat, then M is either a η-Einstein-Sasakian manifold or a (k, μ)-contact manifold with μ = 1 and k = \frac{τ}{2n} − n + 2.

2 Preliminaries

Let M be an m-dimensional Riemannian manifold with metric g and let X (M) be the Lie algebra of differentiable vector fields in M. Denote by R the Riemannian curvature tensor on M. The Ricci operator Q of (M,g) is defined by

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 1-7 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(2.1)  g(QX, Y) = ∑_{i=1}^{m} g(R(e_{i}, X)Y, e_{i}),

where {e_{1}, e_{2}, ..., e_{m}} is a local orthonormal basis of vector fields on M and X, Y ∈ X (M). Weyl introduced a generalized curvature tensor on a Riemannian manifold which vanishes whenever the metric is (locally) conformally equivalent to a flat metric; for this reason he called it the conformal curvature tensor at the metric. Schouten [8] showed that for m > 3 the converse is true. If τ denotes the scalar curvature of M, the Weyl conformal curvature tensor is defined as a map

C : X (M) × X (M) × X (M) → X (M)

such that (2.2)

\begin{aligned}C(X, Y)Z & = & R(X, Y)Z − \frac{1}{m − 2}[g(QY, Z)X + g(Y, Z)QX − g(QX, Z)Y \\ & − & g(X, Z)QY + \frac{τ}{(m − 1)(m − 2)}[g(Y, Z)X − g(X, Z)Y], \end{aligned}

for any X, Y, Z ∈ T (M).

Now let (M, g, φ, ξ, η) be a (2n + 1)-dimensional contact metric manifold (See [2]). Then φ is a (1,1)-tensor field, ξ is the contact vector field, η is the contact 1-form and g is the associated Riemannian metric. They are related by

(2.3)  φ^{2} = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, dη(ξ, X) = 0,
g(φX, φY) = g(X, Y) − η(X)η(Y), η(X) = g(X, ξ), dη(X, Y) = g(X, φY),

for any vector fields X, Y ∈ X (M). Denoting by l and R Lie derivation and the curvature tensor respectively, we define the operators l and h by

(2.4)  lX = R(X, ξ)ξ, hX = \frac{1}{2}(L_{ξ}φ)X.

The (1,1) tensors h and l are self-adjoint and satisfying

(2.5)  hξ = 0, lξ = 0, tr(h) = tr(hφ) = 0, hφ = −φh.

A contact metric manifold for which ξ is a 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. Moreover on a K-contact manifold it is valid R(X, ξ)ξ = X −η(X)ξ. A contact metric manifold is said to be a Sasakian manifold if

(2.6)  (∇_{X}φ)Y = g(X, Y)ξ − η(X)Y.

Note that a Sasakian manifold is K-contact but the converse holds if dimM = 3. A contact metric manifold M is said to be η-Einstein if the Ricci operator satisfies

(2.7)  Q = aI_{d} + bη ⊗ ξ,

for the smooth functions a, b on M [2].

The sectional curvature K(ξ, X) of a plane section spanned by ξ and a vector X orthogonal to ξ is called a ξ-sectional curvature while the sectional curvature K(X,φX) is called a φ-sectional curvature.

Let M be a contact metric manifold. The (k, μ)-nullity distribution of M for the pair (k, μ) is a distribution (2.8)

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

where k, μ ∈ R ([2], and [3]).

So if the characteristic vector field ξ belongs to the (k, μ)-nullity distribution we have:

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

Let C be the Weyl conformal curvature tensor of M. Since at each point p ∈ M the tangent space T_{p}(M) can be decomposed into the direct sum T_{p}(M) = φ(T_{p}(M)) ⊕ L(ξ_{p}), where L(ξ_{p}) is a 1-dimensional linear subspace of T_{p}(M) generated by ξ_{p}, we have a map:

C : T_{p}(M) × T_{p}(M) × T_{p}(M) → φ(T_{p}(M)) ⊕ L(ξ_{p}).

It may be natural to consider the following particular cases:

(1) C : T_{p}(M) × T_{p}(M) × T_{p}(M) → L(ξ_{p}), that is, the projection of the image of C in φ(T_{p}(M)) is zero.

(2) C : T_{p}(M)×T_{p}(M)×T_{p}(M) → φ(T_{p}(M)), that is, the projection of the image of C in L(ξ_{p}) is zero.

(3) C : φ(T_{p}(M)) × φ(T_{p}(M)) × φ(T_{p}(M)) → L(ξ_{p}), that is, when C is restricted to φ(T_{p}(M)) × φ(T_{p}(M)) × φ(T_{p}(M)), the projection of the image of C in φ(T_{p}(M)) is zero [3]. This condition is equivalent to

(2.10)  φ^{2}C(φX, φY)φZ = 0.

A Riemannian manifold satisfying (2.10) is called φ-conformally flat.

The case (1) and (2) are considered in [6] and [7] respectively. The case (3) is considered in [3] for the case M is K-contact.

3 Known results

In this section we give some known results. Proposition 3.1 (2). Let M be a (2n + 1)-dimensional contact metric manifold with

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

for any vector fields X, Y. Then

(3.1)  Qφ − φQ = 2[2(n − 1) + μ]hφ.

Proposition 3.2 (2). Let M be a (k, μ)-contact metric manifold. Then i) R(ξ, X)Y = k[g(X, Y)ξ − η(Y)X] + μ[g(hX, Y)ξ − η(Y)hX), ii) Qξ = 2nkξ.

Lemma 3.3 (2). Let M be a (k, μ)-contact metric manifold, where k < 1. For any vector field X, the Ricci operator Q is given by

QX = [2(n − 1) − nμ]X + [2(n − 1) + μ]hX + [2(1 − n) + n(2k + μ)]η(X)ξ; n ≥ 1.

Theorem 3.4 (2). Let M be a (k, μ)-contact Riemannian manifold. Then k ≤ 1. If k = 1, then h = 0 and M is a Sasakian manifold. If k < 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 − k}.

In the proof of the main theorem we also use the following result. Theorem 3.5 (4). Let M be a (2n + 1)-dimensional (n > 1) non-Sasakian (k, μ)– contact Riemannian manifold. Then M has constant φ-sectional curvature if and only if μ = k + 1.

4 φ-conformally flat contact metric manifolds

In this section we generalize the following theorem; Theorem 4.1 (3). A compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S^{1}-bundle over an almost Kaehler space of constant halomorphic sectional curvature.

First we give Lemma 4.2. Let M be a (2n + 1)-dimensional φ-conformally flat Riemannian manifold. If M is (k, μ)–contact, then

(4.1)  \begin{aligned}S(Y, Z) & = & (\frac{τ}{2n} − k) g(Y, Z) + [2(2(n − 1) + μ)− \\ & − & μ(2n − 1)]g(hY, Z)− \\ & − & (\frac{τ}{2n} − 2nk − k) η(Y)η(Z), (k ≤ 1) \end{aligned}

Proof. It is easy to see that φ^{2}C(φX, φY)φZ = 0 holds if and only if

g(C(φX, φY)φZ, φW) = 0,

for any vector fields X, Y, Z, W ∈ X (M). So φ-conformally flat means

(4.2)  \begin{matrix}g(R(φX, φY)φZ, φW) = \\ = \frac{1}{2n − 1}[g(QφY, φZ)g(φX, φW) + g(φY, φZ)g(QφX, φW)− \\ −g(QφX, φZ)g(φY, φW)-g(φX, φZ)g(QφY, φW)] \\ −\frac{τ}{2n(2n-1)}[g(φY,φZ)g(φX,φW)-g(φX,φZ)g(QφY,φW)]. \end{matrix}

Let {e_{1}, ..., e_{2n}, ξ} be a local orthonormal basis of vector fields in M. By using that {φe_{1},...,φe_{2n},ξ} is also a local orthonormal basis, if we put X = W = e_{α} in (4.2) and sum up with respect to α, then (4.3)

∑_{α=1}^{2n} g(R(φe_{α}, φY)φZ, φe_{α}) = ∑_{α=1}^{2n} \frac{1}{2n − 1}[g(QφY, φZ)g(φe_{α}, φe_{α})

+g(φY, φZ)g(Qφe_{α}, φe_{α}) − g(Qφe_{α}, φZ)g(φY, φe_{α}) − g(φe_{α}, φZ)g(QφY, φe_{α})]

−\frac{τ}{2n(2n − 1)}[g(φY, φZ)g(φe_{α}, φe_{α}) − g(φe_{α}, φZ)g(φY, φe_{α})].

By the use of Proposition 3.2 ii) and the definition of the scalar curvature τ we have

(4.4)  τ − 2nk = ∑_{α=1}^{2n} g(Qφe_{α}, φe_{α}).

Substituting (4.4) into (4.3) we obtain

g(QφY, φZ) − g(R(ξ, φY)φZ, ξ)

= \frac{1}{2n − 1}[2(n − 1)g(QφY, φZ) + (τ − 2nk)g(φY, φZ)]

−\frac{τ}{2n(2n − 1)}[(2n − 1)g(φY, φZ)].

The last equation can be also written as

(4.5)  g(QφY, φZ) − (2n − 1)g(R(ξ, φY)φZ, ξ) = (\frac{τ}{2n} − 2nk)g(φY, φZ)].

Now, since M is (k, μ)-contact, then by Theorem 3.3 we find

(4.6)  g(R(ξ, φY)φZ, ξ)) = kg(φY, φZ) + μg(hφY, φZ).

Using the relations (2.3) (2.5) we can also get

(4.7)  g(hφY, φZ) = −g(hY, Z).

Further, by the use of (3.1) and (2.3) we obtain respectively

(4.8)  g(QφY, φZ) = g(φQY, φZ) + 2[2(n − 1) + μ]g(hφY, φZ)

and

(4.9)  g(φQY, φZ) = g(QY, Z) − 2nkη(Y)η(Z).

Finally, substituting (4.8), (4.7) and (4.6) into (4.5) after some computations we find

g(QY, Z) − 2nkη(Y)η(Z) − 2[2(n − 1) + μ]g(hY, Z)−

−(2n − 1)[k(g(Y, Z) − η(Y)η(Z)) − μg(hY, Z)] =

= (\frac{τ}{2n} − 2nk) [g(Y, Z) − η(Y)η(Z)].

Since S(Y, Z) = g(QY, Z) then the last equality can be written as (4.1). This completes the proof the Lemma.

By Lemma 4.1 we obtain the following result. Theorem 4.3. Let M be a contact metric manifold under the condition that ξ belongs to (k, μ)-nullity condition. If M is φ-conformally flat, then M is either a η-Einstein-Sasakian manifold or a (k, μ)-contact manifold with μ = 1 and k = \frac{τ}{2n} − n + 2. Proof. Since M is φ-conformally flat (k, μ)-contact manifold, then by the previous Lemma the relation (4.1) is satisfied. Also by the use of Theorem 2.3, we have k ≤ 1.

Now, if k = 1 then M is Sasakian and equation (4.1) reduces to

S(Y, Z) = (\frac{τ}{2n} − 1)g(Y, Z) − (\frac{τ}{2n} − 2n − 1)η(Y)η(Z).

So by the use of (2.1) and (2.6) it is easy to show that M is a η-Einstein manifold. If k < 1, then by Lemma 3.2 we find

(4.10)  \begin{aligned}S(Y, Z) & = & [2(n − 1) − nμ]g(Y, Z) + [2(n − 1) + μ]g(hY, Z) \\ & + & [2(1 − n) + n(2k + μ)]η(Y)η(Z). \end{aligned}

However, comparing the equations (4.9) and (4.1) we obtain the following system of equations:

\frac{τ}{2n} − k = 2(n − 1) − nμ,

2(2(n − 1) + μ) − μ(2n − 1) = 2(n − 1) + μ,

− (\frac{τ}{2n} − 2nk − k) = 2(1 − n) + n(2k + μ).

Solving this system we get μ = 1, k = \frac{τ}{2n} − n + 2. Hence by the virtue of Theorem 3.3 M must have constant φ-sectional curvature. This completes the proof of the theorem.

By Theorem 4.3 we have the following result. Corollary. Let M be a (0, μ)-contact manifold. If M is φ-conformally flat manifold, then M has constant φ-sectional curvature. Proof. Let M be a (0, μ)-contact φ-conformally flat manifold then by Theorem 4.3 μ = 1. Hence by the virtue of Theorem 3.3 M must have constant φ-sectional curvature.

References

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Uludag University

Faculty of Art and Sciences

Department of Mathematics, 16059 Gorukle/Bursa/TURKEY E mail: arslan@@uludag.edu.tr