Math. J. Okayama Univ. 57 (2015), 149–158

ON φ-RECURRENT CONTACT METRIC MANIFOLDS
E. Peyghan, H. Nasrabadi and A. Tayebi
Abstract. In this paper, we prove that evry 3-dimensional manifold
M is a φ-recurrent N (k)-contact metric manifold if and only if it is flat.
Then we classify the φ-recurrent contact metric manifolds of constantcurvature. This implies that there exists no φ-recurrent N (k)-contact
metric manifold, which is neither symmetric nor locally φ-symmetric.

1. Introduction
In 1872, S. Lie introduced the notion of contact transformation as a geometric tool to study systems of differential equations [1][2][3]. The theory ofcontact metric structures occupies one of the leading places in researches ofmodern differential geometry because of its several applications in mechanics, optics, phase space of a dynamical system, control theory and in thetheory of geometrical quantization [11].
On the other hand, the internal contents of the theory of contact metricstructures are rich and have close substantial interactions with other partsof geometry. For example, Sasakian manifolds play important role in contact geometry. Indeed, the links between contact geometry and complex
geometry are especially strong for Sasakian manifolds [3][5][7].
It is shown that, the only locally symmetric Sasakian manifolds are locallyisometric to S 2n+1 (1) and that the only locally symmetric contact metricmanifolds are locally isometric to S 2n+1 (1) or to E n+1 × S n (4) (see [3]).
Certainly this can be regarded as saying that the idea of being locally symmetric is too strong. For this reason, this notion has been weakend by manygeometers in different ways such as recurrent manifold by Walker [17], semi
symmetric manifold by Szabó [14], pseudo-symmetric manifold by Chaki[6], and Deszcz [10] and weakly symmetric manifold by Tammasy and Binh[16], and Selberg [13]. As a weaker version of local symmetry, Takahashi
introduced the notion of a locally φ-symmetric space [15]. Generalizing thenotion of local φ-symmetry, De-Shaikh-Biswas introduced the notion of φrecurrent Sasakian manifold [9]. Then in [8], De-Gazi studied φ-recurrent
N (k)-contact metric manifolds and generalized the results of [9].
In [8], De-Gazi proved that a 3-dimensional φ-recurrent N (k)-contact
metric manifold is of constant curvature. Then they provided the existence
Mathematics Subject Classification. Primary 53C15; Secondary 53C40.
Key words and phrases. Constant curvature, Locally φ-symmetric, N (k)-contact metricmanifold, φ-recurrent.
149

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E. PEYGHAN, H. NASRABADI AND A. TAYEBIof the φ-recurrent N (k)-contact metric manifold by means of an example
which is neither symmetric nor locally φ-symmetric.
This paper is arranged as follows. In Section 2, we present some basic
concepts of the contact metric manifolds, Sasakian manifolds, locally φsymmetric manifolds and φ-recurrent N (k)-contact metric manifolds. In
Section 3, we show that the example introduced by De-Gazi in [8] is notcorrect. Then, we prove that a 3-dimensional manifold M is φ-recurrent
N (k)-contact metric manifold if and only if it is a flat manifold. In otherwords, we prove that there exists no 3-dimensional φ-recurrent N (k)-contact
metric manifold for k 6= 0. We also deduce that there exists no 3-dimensional
φ-recurrent N (k)-contact metric manifold which is neither symmetric nor
locally φ-symmetric. All results in this section show that Theorem 4.1 in [8]is not correct. In Section 4, we show that for k 6= 0, there is no (2n + 1)dimensional φ-recurrent N (k)-contact metric manifold of contact curvature.
We also prove that there is no (2n + 1)-dimensional φ-recurrent contact
metric manifold of contact curvature for n > 1. Finally we show that only,
the flat 3-dimensional manifolds are φ-recurrent contact metric manifold ofconstant curvature.
2. Contact Metric Manifolds
In this section, we remark some fundamental materials about contact
metric geometry. We refer to [3], [12] for further details.
A (2n + 1)-dimensional manifold M 2n+1 is said to be a contact manifold
if it admits a global 1-form η such that η ∧ (dη)n 6= 0, everywhere. Given a
contact form η, there exists a unique vector field ξ, the characteristic vector
field, which satisfies η(ξ) = 1 and dη(ξ, X) = 0 for any vector field X. It is
well known that, there exists an associated Riemannian metric g and a (1,1)-type tensor field φ such that the following hold
(2.1)(i) η(X) = g(X, ξ), (ii) dη(X, Y ) = g(X, φY ), (iii) φ2 X = −X + η(X)ξ,where X and Y are vector fields on M . By (2.1), it follows thatφξ = 0,η ◦ φ = 0,g(φX, φY ) = g(X, Y ) − η(X)η(Y ).

A Riemannian manifold M equipped with the structure tensors (η, ξ, φ, g)satisfying (2.1) is said to be a contact metric manifold.
Given a contact metric manifold M , we define a (1, 1)-tensor field h by
h = 21 £ξ φ, where £ denotes the Lie differentiation. Then the tensor h issymmetric and satisfies(2.2)(i) hξ = 0, (ii) hφ + φh = 0, (iii) ∇X ξ = −φX − φhX,

2(∇hX φ)Y = −R(ξ, X)Y − φR(ξ, X)φY + φR(ξ, φX)Y − R(ξ, φX)φYON φ-RECURRENT CONTACT METRIC MANIFOLDS

151+ 2g(X + hX, Y )ξ − 2η(Y )(X + hX),
where ∇ is the Levi-Civita connection and R is the Riemannian curvaturetensor of M defined by following
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,

∀X, Y, Z ∈ χ(M ).

For a contact metric manifold M , one may defines naturally an almost complex structure J on M × R as followsddJ(X, f ) = (φX − f ξ, η(X) ),dt
dt
where X is a vector field tangent to M , t the coordinate on R and f afunction on M × R. If the almost complex structure J is integrable, M issaid to be normal or Sasakian. It is known that, a contact metric manifold
M is normal if and only if M satisfies[φ, φ] + 2dη ⊗ ξ = 0,
where [φ, φ] is the Nijenhuis torsion of φ. It is also well known that a contact
metric manifold M is Sasakian if and only if
R(X, Y )ξ = η(Y )X − η(X)Y,∀X, Y ∈ χ(M ).

The k-nullity distribution N (k) of a Riemannian manifold M is definedbyno
N (k) : p −→ Np (k) = Z ∈ Tp M : R(X, Y )Z = k[g(Y, Z)X − g(X, Z)Y ] ,where k is a constant. If the characteristic vector field ξ belongs to N (k),then we call a contact metric manifold an N (k)-contact metric manifold. If
k = 1, then N (k)-contact metric manifold is Sasakian and if k = 0, thenN (k)-contact metric manifold is locally isometric to the product E n+1 ×
S n (4) for n > 1 and flat for n = 1 (see [3]). For a N (k)-contact metricmanifold we have
(2.3)(i) R(X, Y )ξ = k[η(Y )X − η(X)Y ],(ii) S(X, ξ) = 2nkη(X),where S is the Ricci tensor of Riemannian manifold (M, g) (see [8]).
A contact metric manifold is said to be locally φ-symmetric if the relation
φ2 ((∇W R)(X, Y )Z) = 0,
holds for all vector fields X, Y , Z, W orthogonal to ξ [4]. This notion was
introduced for Sasakian manifolds by Takahashi [15].
Definition 2.1. A contact metric manifold is said to be φ-recurrent if there
exists a non-zero 1-form A such that(2.4)

φ2 ((∇W R)(X, Y )Z) = A(W )R(X, Y )Z,

for all vector fields X, Y, Z, W .

152

E. PEYGHAN, H. NASRABADI AND A. TAYEBIIn the above definition, X, Y, Z, W are arbitrary vector fields and not necessarily orthogonal to ξ. This notion was introduced for Sasakian manifolds
by De, Shaikh and Biswas [9] and was introduced for N (k)-contact manifolds
by De and Gazi [8].
Remark 2.2. Flat manifolds are trivial examples of φ-recurrent contact
metric manifolds (locally φ-symmetric manifolds), because for a flat manifold
we have R = 0 and ∇R = 0.
3. 3-dimensional φ-recurrent N (k)-contact metric manifolds
In [8], De-Gazi presented the following example of φ-recurrent N (k)contact metric manifold which is neither symmetric nor locally φ-symmetric.
Example 3.1. We take the 3-dimensional manifold M = {(x, y, z)|x 6= 0},
where (x, y, z) are the standard coordinates in R3 . Let E1 , E2 , E3 be linearly
independent global frame on M given by
E1 =

∂
4z ∂
∂
∂
2 ∂
, E2 = 2−+ xy , E3 =
.
x ∂y
∂x
x ∂y
∂z
∂z

Let g be the Riemannian metric defined by g(E1 , E3 ) = g(E2 , E3 ) = g(E1 , E2 ) =0 and g(E1 , E1 ) = g(E2 , E2 ) = g(E3 , E3 ) = 0. Let η be the 1-form defined by η(U ) = g(U, E3) for any U ∈ χ(M ). Let φ be the (1, 1) tensor field defined by φE1 = E2 , φE2 = −E1 , φE3 = 0. Then using thelinearity of φ and g we have η(E3) = 1, φ2 (U ) = −U + η(U )E3 andg(φU, φW ) = g(U, W ) − η(U )η(W ) for any U, W ∈ χ(M ). Moreover hE1 =−E1 , hE2 = E2 , hE3 = 0. Thus for E3 = ξ, (φ, ξ, η, g) defines a contact
metric structure on M . Hence we have [E1 , E2 ] = 2E3 + x2 E1 , [E1 , E3 ] = 0,
[E2 , E3 ] = 2E1 .
The Riemannian connection ∇ of the metric g is given by
2g(∇X Y, Z) = Xg(Y, Z) + Y g(Z, X) − Zg(X, Y )− g(X, [Y, Z]) − g(Y, [X, Z]) + g(Z, [X, Y ]).
Taking E3 = ξ and using the above formula for Riemannian metric g, it canbe easily calculated that2E1x
2
∇E2 E1 = −2E3 , ∇E2 E2 = 0, ∇E3 E2 = 0, ∇E1 E1 = − E2 .
x
∇E1 E3 = 0, ∇E2 E3 = 2E1 , ∇E3 E3 = 0, ∇E1 E2 =(3.1)

From the above it can be easily seen that (φ, ξ, η, g) is a N (k)-contact metricmanifold with k = − x4 6= 0.

ON φ-RECURRENT CONTACT METRIC MANIFOLDS

153

Now, we are going to show that the above example is not correct. By
using (3.1), we get
R(E1 , E2 )E3 = ∇E1 ∇E2 E3 − ∇E2 ∇E1 E3 − ∇[E1,E2 ] E32
= 2∇E1 E1 − 2∇E3 E3 − ∇E1 E3
x
4
(3.2)
= − E2 .
x
On the other hand, since η(E1 ) = g(E1 , E3 ) = 0 and η(E2 ) = g(E2 , E3 ) = 0,then we have
(3.3)
R(E1 , E2 )E3 = k η(E2 )E1 − η(E1 )E2 = 0,
But (3.2) contradicts with (3.3).

In [8], the authors proved the following theorem (See Theorem 4.1 in [8]).
Theorem 3.2. ([8]) Every 3-dimensional φ-recurrent N (k)-contact metricmanifold is of constant curvature.
On the other hand, Blair proved the following.
Theorem 3.3. ([3]) A contact metric manifold M 2n+1 satisfying R(X, Y )ξ =0 is locally isometric to E n+1 × S n (4) for n > 1 and flat for n = 1.
Remark 3.4. Using the above theorem, if k = 0, then N (k)-contact metricmanifold M 3 is flat. Thus according to Remark 2.2, it is easy to see that 3dimensional N (k)-contact metric manifold M is φ-recurrent, symmetric andlocally φ-symmetric. Therefore according to the Example 3.1, it is deduced
that the Theorem 3.2 is proved for k 6= 0.
Here, we show that Theorem 3.2 (Theorem 4.1 in [8]) is, strictly speaking,not correct. At first, we prove the following theorem:
Theorem 3.5. Every 3-dimensional manifold M is a φ-recurrent N (k)contact metric manifold if and only if it is a flat manifold.
Proof. It is known that the Riemannian curvature of a 3-dimensional Riemannian manifold M satisfies inR(X, Y )Z = g(Y, Z)QX − g(X, Z)QY + S(Y, Z)X − S(X, Z)Yr(3.4)+ [g(X, Z)Y − g(Y, Z)X],
2
where Q is the Ricci operator, that is , g(QX, Y ) = S(X, Y ) and r is thescalar curvature of M . Now, let M be a 3-dimensional φ-recurrent N (k)contact metric manifold. Putting Z = ξ in (3.4) and using (ii) of (2.3) and154

E. PEYGHAN, H. NASRABADI AND A. TAYEBI

η(ξ) = 1, yieldsrR(X, Y )ξ = (2k − )[η(Y )X − η(X)Y ] + η(Y )QX − η(X)QY.
2Part (i) of (2.3) and (3.5) give usr(3.6)(k − )[η(Y )X − η(X)Y ] = η(X)QY − η(Y )QX.
2
By setting Y = ξ in (3.6) and using (ii) of (2.3), it follows that
r
r
(3.7)
QX = ( − k)X + (3k − )η(X)ξ,22
which gives us
r
r
(3.8)
S(X, Y ) = g(QX, Y ) = ( − k)g(X, Y ) + (3k − )η(X)η(Y ).
2
2
Using (3.7), (3.8) and (3.4) we getrR(X, Y )Z = (3k − )[g(Y, Z)η(X)ξ − g(X, Z)η(Y )ξ + η(Y )η(Z)X
2
r
(3.9)
− η(X)η(Z)Y ] + ( − 2k)[g(Y, Z)X − g(X, Z)Y ].
2
By (3.9), we get(3.5)(∇W R)(X, Y )Z = ∇W R(X, Y )Z − R(∇W X, Y )Z
− R(X, ∇W Y )Z − R(X, Y )∇W Z
dr(W )[g(Y, Z)X − g(X, Z)Y − g(Y, Z)η(X)ξ
2+ g(X, Z)η(Y )ξ − η(Y )η(Z)X + η(X)η(Z)Y ]r+ (3k − )[g(Y, Z)η(X) − g(X, Z)η(Y )]∇W ξ
2r+ (3k − )[η(Y )X − η(X)Y ](∇W η)(Z)2r+ (3k − )[g(Y, Z)ξ − η(Z)Y ](∇W η)(X)2r− (3k − )[g(X, Z)ξ − η(Z)X](∇W η)(Y ).
(3.10)
2
Now, let Y be a non-zero vector field orthogonal to ξ and X = Z = ξ. Thenfrom (3.10), we getr(3.11)(∇W R)(ξ, Y )ξ = −2(3k − )(∇W η)(ξ)Y.
2
Since η(ξ) = 1 and η ◦ φ = 0, then using part (iii) of (2.2) we obtain=(3.12)(∇W η)(ξ) = W (η(ξ)) − η(∇W ξ) = η(φW + φhW ) = 0.

ON φ-RECURRENT CONTACT METRIC MANIFOLDS

155

By plugging (3.12) in (3.11), we have
(3.13)(∇W R)(ξ, Y )ξ = 0.

Since M is a φ-recurrent manifold, then there exists a non-zero 1-form A
such that satisfies in (2.4). Thus using (2.4) and (3.13), we deduce that
(3.14)

A(W )R(ξ, Y )ξ = 0.

As M is N (k)-contact metric manifold, then we have
(3.15)

R(ξ, Y )ξ = k[η(Y )ξ − η(ξ)Y ] = −kY.

Setting (3.15) in (3.14), implies that kA(W )Y = 0 which gives us k = 0.
Thus we have R(X, Y )ξ = 0. Therefore, by using the Theorem 3.3 we canconclude that M 3 is a flat manifold. From Remark 3.4, the converse of thetheorem is obvious.

Using Remark 3.4 and the Theorem 3.5, we have the following.
Corollary 3.6. There is no φ-recurrent N (k)-contact metric manifold withdimension 3, for k 6= 0.
From the above corollary we deduce
Corollary 3.7. There is no 3-dimensional φ-recurrent Sasakian manifold.
Using Remark 3.4, Theorem 3.5 and Corollary 3.6, we conclude the following.
Theorem 3.8. There is no 3-dimensional φ-recurrent N (k)-contact metricmanifold which is neither symmetric nor locally φ-symmetric
Therefore, by using Remark 3.4, Corollary 3.6 and Theorem 3.8, we conclude that Theorem 3.2 is not correct.
4. φ-recurrent contact metric manifolds of constantcurvature
In this section, we show that the only flat manifolds of dimension 3 can be
φ-recurrent contact metric manifolds of constant curvature. For this work,
we present a fact that shows Theorem 3.2 (Theorem 4.1 in [8]) is, strictly
speaking, not correct, again.
Theorem 4.1. For k 6= 0, there exists no (2n + 1)-dimensional φ-recurrent
N (k)-contact metric manifold of constant curvature.
Proof. Let M 2n+1 be a φ-recurrent N (k)-contact metric manifold which has
the constant curvature λ. Then we have
(4.1)

R(X, Y )Z = λ(g(Y, Z)X − g(X, Z)Y ).

156

E. PEYGHAN, H. NASRABADI AND A. TAYEBI

Setting Z = ξ in (4.1) it yields(4.2)

R(X, Y )ξ = λ(η(Y )X − η(X)Y ).

Since M is N (k)-contact metric manifold, then we have
(4.3)

R(X, Y )ξ = k(η(Y )X − η(X)Y ).

(4.2) and (4.3) give us(4.4)

k(η(Y )X − η(X)Y ) = λ(η(Y )X − η(X)Y ).

Let X = ξ and Y be a non-zero vector field orthogonal to ξ. Then we haveη(Y ) = 0 and η(X) = 1. Thus from (4.4), we deduce thatkY = λY,orλ = k 6= 0.

Thus by using (4.1), we have
(4.5)

R(X, Y )Z = k(g(Y, Z)X − g(X, Z)Y ).

By (4.5), it follows that(∇W R)(X, Y )Z = ∇W R(X, Y )Z − R(∇W X, Y )Z − R(X, ∇W Y )Z
− R(X, Y )∇W Z = −k[(∇W g(X, Z))Y + g(X, Z)∇W Y− (∇W g(Y, Z))X − g(Y, Z)∇W X − g(∇W X, Z)Y+ g(Y, Z)∇W X − g(X, Z)∇W Y + g(∇W Y, Z)X− g(X, ∇W Z)Y + g(Y, ∇W Z)X]
= −k[((∇W g)(X, Z))Y − ((∇W g)(Y, Z))X]
(4.6)

= 0.

Also from (4.5), we get(4.7)

R(X, ξ)ξ = k[X − η(X)ξ].

Putting Y = Z = ξ in (2.4) and using (4.6) and (4.7) imply that
(4.8)

kA(W )[X − η(X)ξ] = 0.

If X is a non-zero vector field orthogonal to ξ, then (4.8) gives us kA(W )X =0, which is a contradiction to k 6= 0 and A(W ) 6= 0.
In [3], Blair proved the following.
Theorem 4.2. If a contact metric manifold M 2n+1 is of constant curvature
λ and n > 1, then λ = 1 and the structure is Sasakian.
Here, we are going to consider the same result for φ-recurrent contact
metric manifold. Then we prove the following.
Theorem 4.3. there exists no (2n + 1)-dimensional φ-recurrent contact
metric manifold of constant curvature in which n > 1.

ON φ-RECURRENT CONTACT METRIC MANIFOLDS

157

Proof. Let M be a φ-recurrent contact metric manifold of constant curvature
λ with dimension 2n + 1, where n > 1. Considering Theorem 4.2, we deducethat λ = 1. Thus we have
(4.9)

R(X, Y )Z = g(Y, Z)X − g(X, Z)Y.

In a similar way to the proof of (4.6) in Theorem 4.1, from (4.9) we get thefollowing(4.10)(∇W R)(X, Y )Z = 0.

Let X = Z = ξ and Y be a non-zero vector field orthogonal to ξ. Then (4.9)
gives us(4.11)

R(ξ, Y )ξ = Y.

Since M is φ-recurrent, then by using (2.4), (4.10) and (4.11) we deducethat
A(W )Y = 0.
But this contradicts A(W ) 6= 0.
Theorem 4.4. A φ-recurrent contact manifold M 3 is of constant curvature
λ if and only if λ = 0.
Proof. Let M be a 3-dimensional φ-recurrent contact manifold of constantcurvature λ. Then we have
R(X, Y )Z = λ(g(Y, Z)X − g(X, Z)Y ).
Since M 3 is φ-recurrent then similar to proof of Theorem 4.3, by using theabove equation, we obtain λA(W )Y = 0, where A is a non zero 1-form andY is a non zero vector field on M . Thus we deduce that λ = 0. According
to Remarks 2.2 and 3.4, the converse of the theorem is obvious.

Theorems 4.3 and 4.4 give us the following.
Corollary 4.5. The only flat 3-dimensional φ-recurrent contact metric manifolds, are manifolds of constant curvature.
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Esmaeil Peyghan
Department of Mathematics, Faculty of Science
Arak University
Arak 38156-8-8349, Irane-mail address: e-peyghan@araku.ac.ir
Hassan Nasrabadi
Department of Mathematics, Faculty of Science
Arak University
Arak 38156-8-8349, Iran
Akbar Tayebi
Department of Mathematics, Faculty of Science
University of Qom
Qom. Irane-mail address: akbar.tayebi@gmail.com(Received May 13, 2013 )(Revised September 11, 2013 )