EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 5, 2017, 946-954
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Harmonic and (φ, φ́)-holomorphic maps on some
generalization of contact metric manifolds

Fereshteh Malek1, Mahboobeh Samanipour2,∗

1 Faculty of Mathematics, K. N. Toosi University of Technology,Tehran, Iran
2 Department of Mathematics, Payame Noor University, P.O.Box.19395-3697, Tehran, Iran

Abstract. We study (φ, φ́)-holomorphic and harmonic maps on almost contact metric manifolds
satisfying dη + dηoφ ⊗ φ = 2Φ. It should be noted that these manifolds are not necessarily
contact metric manifolds. Examples of such manifolds are nearly Sasakain and quasi contact
metric manifolds.

2010 Mathematics Subject Classifications: 53C40, 53C15, 53C25

Key Words and Phrases: Quasi contact metric manifold, Nearly Sasakian manifold, Quasi
Kähler manifold, Holomorphic map, Contact metric manifold

1. Introduction

J. Eells and J. H. Sampson studied harmonic maps on Riemannian manifolds ([8]).
Also, they proved that a holomorphic map between Kähler manifolds is a harmonic map. In
odd dimension, the almost contact metric manifolds represent the analogue of almost Her-
mitian manifolds (see [2], [3]). S. Ianus and A. M. Pastore, considered (φ, φ́)-holomorphic
maps between contact metric (or Sasakian) manifolds and proved that such maps are har-
monic ([9]). Contact metric manifolds are (φ, ξ, η, g)- almost contact metric manifolds in
which dη = Φ.

In this paper, we study (φ, φ́)-holomorphic maps between two classes of almost contact
metric manifolds in which dη+ dηoφ⊗φ = 2Φ. Nearly Sasakian and quasi contact metric
manifolds are examples of these kind of manifolds (contact metric manifolds, satisfy this
relation too, because in contact metric manifolds we have dηoφ⊗φ = dη). Also, conformal
transformation on these manifolds are considered in this paper. In [6], Tanno studied
conformal transformation on contact metric manifolds.

∗Corresponding author.

Email addresses: malek@kntu.ac.ir (F. Malek), samani108@yahoo.com (M. Samanipour)

http://www.ejpam.com 946 c© 2017 EJPAM All rights reserved.



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 947

2. Preliminaries

A contact manifold is a C∞ manifold M2n+1 together with a 1-form η such that η ∧
(dη)n 6= 0. In a contact manifold (M,η) we can choose a vector field ξ, called characteristic
vector field, such that dη(ξ,X) = 0 and η(ξ) = 1.

We say that a Riemannian manifold (M, g) has an almost contact metric structure,
if it admits a tensor field φ of type (1,1), a vector field ξ and a 1-form η satisfying the
following relations

φ2 = −I + η ⊗ ξ, g(X, ξ) = η(X), g(φX, φY ) = g(X,Y )− η(X)η(Y ).

It is proved that given a contact manifold (M,η) with characteristic vector field ξ,
there exists an almost contact metric structure(φ, ξ, η, g) such that dη(X,Y ) = Φ(X,Y ),
in which Φ(X,Y ) = g(X,φY ). In this case, we say that M is equipped with a (φ, ξ, η, g)
contact metric structure. We mention that it is possible to have a contact manifold
M = (M,η) with characteristic vector field ξ and with (φ, ξ, η, g)-almost contact metric
structure , same ξ and η , in which dη(X,Y ) 6= g(X,φY ), then M = (M,φ, ξ, η, g) is
contact but not contact metric.

Y. Tashiro studied almost contact manifolds as hypersurfaces of almost complex man-
ifolds ([16]). He showed that a hypersurface in an almost Hermitian manifold has an
almost contact metric structure and when the almost Hermitian manifold is quasi Kähler,
he called it’s associated almost contact metric hypersurface, contact O∗−manifold. Then
J. H. Kim, J. H. Park and K. Sekigawa called such an almost contact metric manifold a
quasi contact metric manifold ([10]). They proved that an almost contact metric manifold
M = (M,φ, ξ, η, g) is quasi contact metric if and only if it satisfies the following relation:

(∇Xφ)Y + (∇φXφ)φY = 2g(X,Y )ξ − η(Y )(X + η(X)ξ + hX), (1)

for every vector fields X and Y on M, in which h := 1
2Lξφ.

The above relation holds in every contact metric manifold ([3], page116), thus quasi
contact metric manifolds can be regarded as a generalization of contact metric manifolds.

An almost contact metric structure (φ, ξ, η, g) is said to be nearly Sasakian iff

(∇Xφ)Y + (∇Y φ)X = 2g(X,Y )ξ − η(X)Y − η(Y )X (2)

for every vector fields X,Y on M .
Nearly Sasakian manifolds were introduced in 1976 by Blair and his collaborators[4].

They proved that every Sasakian manifold is nearly Sasakian, but the converse state-
ment fails in general. In addition they proved that a normal nearly Sasakian structure
is Sasakian. They also showed that a hypersurface of a nearly kähler manifold is nearly
Sasakian if and only if it is quasi-umbilical with respect to the almost contact form. In [5]



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 948

it is proved that nearly Sasakian manifolds are contact. In [1], [4], [7], [11], [12], [13] and
[15], some other authors studied nearly Sasakian manifolds.

In this paper, we study (φ, φ́)-holomorphic maps between two quasi contact metric (or
nearly Sasakain) manifolds. They are two classes of almost contact metric manifolds in
which dη + dηoφ⊗ φ = 2Φ.

Conformal transformation, holomorphic map and harmonic map have many applica-
tions in physical problems, medical physics, engineering and applied science. Regarding to
the importance of these maps, in this study, conformal transformation, holomorphic map
and harmonic map on these manifolds are considered.

3. Some results on quasi contact metric and nearly Sasakian manifolds

There are some important properties on quasi contact metric and nearly Sasakian
manifolds which we need for the next section.

In the following lemma we list some basic properties of quasi contact metric manifolds
which can be easily proved by using (1) (some of them are proved in [3] and [10]).

Lemma 1. In a quasi contact metric manifold M = (M,φ, ξ, η, g) the following properties
are hold

(a)∇ξφ = 0

(b)∇ξξ = 0

(c)ηoh = 0, hξ = 0

(d)(∇Xη)Y = g(X + hX, φY )

(e)φh+ hφ = 0.

We know that in a (φ, ξ, η, g) contact metric manifold, it satisfies dη = Φ. It should be
very useful if we have an expression for dη with respect to Φ in quasi contact metric and
nearly Sasakian manifolds. The following theorem gives the desired formula, and then we
give a number of important properties on these manifolds by using this formula.

Theorem 1. In a quasi contact metric manifold M = (M,φ, ξ, η, g) we have

dη(X,Y ) = Φ(X,Y ) +
1

2
[g(hX, φY )− g(φX, hY )].

Proof. Using Lemma 1(d), we have

dη(X,Y ) =
1

2
[(∇Xη)Y − (∇Y η)X]



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 949

=
1

2
[g(X + hX, φY )− g(Y + hY, φX)]

= g(X,φY ) +
1

2
[g(hX, φY )− g(φX, hY )].

It is remarkable to note that the above theorem also shows that if h is symmetric, then
the quasi contact metric manifold is contact metric.

In nearly Sasakian manifold (M,φ, ξ, η, g), a tensor field h́ of type (1, 1) is defined by
([5]):

h́X := ∇Xξ + φX.

It is skew-symmetric and anticommutes with φ. Moreover, h́ξ = 0 and ηoh́ = 0 ([5]),
and the vanishing of h́ causes the manifold to be Sasakian ([13]).

Theorem 2. In a nearly Sasakian manifold M = (M,φ, ξ, η, g), we have

dη(X,Y ) = Φ(X,Y ) + g(h́X, Y ).

Proof. We have

dη(X,Y ) =
1

2
[(∇Xη)Y − (∇Y η)X]

=
1

2
[g(∇Xξ, Y )− g(∇Y ξ,X)]

=
1

2
[g(−φX + h́X, Y )− g(−φY + h́ Y,X)]

= Φ(X,Y ) + g(h́X, Y ).

Lemma 2. In a quasi contact metric (or nearly Sasakian) manifold M = (M,φ, ξ, η, g)
we have

dη(φX, φY ) + dη(X,Y ) = 2Φ(X,Y ).

Proof. By Theorem 1 we have:

dη(φX, φY ) + dη(X,Y ) = Φ(φX, φY ) +
1

2
[Φ(hφX, φY ) + Φ(φX, hφY )]

+Φ(X,Y ) +
1

2
[Φ(hX, Y ) + Φ(X,hY )]

= 2Φ(X,Y ) +
1

2
[−Φ(hX, Y )− Φ(X,hY )

+Φ(hX, Y ) + Φ(X,hY )]
= 2Φ(X,Y ).

(By Theorem 2 and applying the same technique the result for nearly Sasakian manifolds
follows.)

Theorems 1 and 2, show that dη(X, ξ) = 0, and then we deduce:



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 950

Corollary 1. In a (φ, ξ, η, g)- quasi contact metric (or nearly Sasakian) manifolds, ξ ∈
kerdη.

Theorem 3. In a quasi contact metric (or nearly Sasakian) manifold M = (M,φ, ξ, η, g),
η is a contact form.

Proof. Let M = (M,φ, ξ, η, g) be a (2n+1)-dimensional quasi contact metric (or nearly
Sasakain) manifold. We prove that rankdη = 2n.

Let p ∈ M and X⊥ξp be a nonzero vector in TpM . If X ∈ kerdη then dη(X, .) = 0.
Thus by Lemma 3.4, we have

0 = dη(X,φX) = Φ(X,φX) = −|X|2,

which is a contradiction. Thus dim(kerdη) ≤ 1. Also by Corollary 1 we have, ξ ∈ kerdη,
and then dim(kerdη) = 1, that results rankdη = 2n. Thus η ∧ (dη)n 6= 0 on TpM and the
proof is completed.

4. Harmonic and (φ, φ́)-holomorphic maps on quasi contact metric, and
nearly Sasakian manifolds

It is known that a contact manifold with contact form η carries an associated almost
contact metric structure (φ, ξ, η, g) with Φ = dη , called a contact metric structure, but
in this section, we consider almost contact metric manifolds in which dη+ dηoφ⊗ φ = 2Φ
and treat conformal transfomations and holomorphic maps on these kind of manifolds.

Let (M, g,∇) and (M,́ ǵ, ∇́) be two Riemannian manifolds in which ∇ and ∇́ are the
Levi-Civita connections on M and M´respectively. For a differentiable map f : M → M´
between Riemannian manifolds its tension field τ(f) is the trace of the second fundamental
form Bf of f :

τ(f) := ΣiBf (ei, ei) = Σi(∇f
−1TḾ
ei df(ei)− df(∇eiei)), (3)

where {ei} is an orthonormal basis for the tangent space TxM at x ∈M , and ∇f−1TḾ

denotes the pull-back of the Levi-Civita connection ∇́ on M´ to the pull-back bundle
f−1TM´−→M , and df : TM −→ f−1TM´is the differential of f .

Definition 1. ([8]) A differentiable map f : M → M´between Riemannian manifolds M
and M´ is called a harmonic map if τ(f) = 0.

Definition 2. ([8]) Let M = (M,φ, ξ, η, g) and M´= (M,́ φ́, ξ́, ή, ǵ) be two almost contact
metric manifolds. We say that a differentiable map f : M → M´ is (φ, φ́)-holomorphic if
dfoφ = φ́odf and f is (φ, φ́)-antiholomorphic if dfoφ = −φ́odf .



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 951

Proposition 1. Let M = (M,φ, ξ, η, g) be a quasi contact metric (or nearly Sasakain)
manifold. If a transformation µ on M leaves φ invariant that means φµ = µφ, then there
exists α ∈ R such that the following relations are hold:
(a) µ∗η = αη
(b) µξ = αξ
(c) µ∗Φ = αΦ

Proof. First we prove (a) and (b) for some α ∈ C∞(M), then we prove that α must
be constant.
(a) From ηoφ = 0 and φµ = µφ, we have ηoµφ = 0. Thus at any point x of M we have:

(µ∗η)(φX)x = 0, X ∈ χ(M).

So,
(µ∗η)x = α(x)ηx,

for some α ∈ C∞(M).
(b) From φξ = 0 and φµ = µφ, we have φµξ = 0. So,

µξ = βξ,

for some β ∈ C∞(M). Therefore η(µξ) = η(βξ) and then (µ∗η)ξ = βη(ξ). By (a) it follows
that α = β.

Now, we show that α must be constant. By differentiation (a), we get:

dµ∗η = dα ∧ η + αdη. (4)

Since d and µ∗ commute and by Corollary 1 and (4), we have:

0 = dη(αξ, µξ)
= dη(µξ, µY )
= µ∗dη(ξ, Y )
= dµ∗η(ξ, Y )
= dα ∧ η(ξ, Y ) + αdη(ξ, Y )
= dα(ξ)η(Y )− dα(Y ), Y ∈ χ(M).

So,
dα(ξ)η = dα. (5)

Therefore, dα(ξ)η∧η = dα∧η. Then, we have dα∧η = 0, and by differentation, we obtain
dα∧ dη = 0. So (5) implies dα(ξ)η ∧ dη = 0 and by Theorem 3 , it follows that dα(ξ) = 0.
Again, condition (5) gives dα = 0, i.e, α is constant.



F. Malek, M. Samanipour / Eur. J. Pure Appl. Math, 10 (5) (2017), 946-954 952

(c) Now by (4), it follows that µ∗dη = αdη. By lemma 2 (2Φ = dη + dη o φ ⊗ φ) we
have:

(µ∗Φ) =
1

2
(µ∗dη + µ∗dη o φ⊗ φ)

=
α

2
(dη + dη o φ⊗ φ)

= αΦ.

In ([6]), Tanno studied conformal transformation on contact metric manifold. In the
following theorem, the same result is proved for quasi contact metric and nearly Sasakain
manifolds.

Theorem 4. Let M = (M,φ, ξ, η, g) be a quasi contact metric (or nearly Sasakain) man-
ifold. If a transformation µ on M leaves φ invariant, then µ is a homothety ( that means
µ∗g = c2g for some nonzero scalar c ) on Γ(D) in which D = {X ∈ TM ; η(X) = 0}).

Proof. For an arbitrary x ∈M and X,Y ∈ Γ(D). We have:

(µ∗Φ)x(X,Y ) = Φ(µX, µY ).

So,
αΦx(X,Y ) = g(µX, µφY ) = (µ∗g)x(X,φY ).

Thus we have
αgx(X,φY ) = (µ∗g)x(X,φY ).

Substituting Y by −φX and we get (µ∗g)(X,X) = αg(X,X), then α > 0.

Theorem 5. Let M = (M,φ, ξ, η, g) and M´= (M,́ φ́, ξ́, ή, ǵ) be two quasi contact metric
manifolds. Any (φ, φ́)-holomorphic (or (φ, φ́)-antiholomorphic) submersion f : M → M´
is a harmonic map.

Proof. Let ∇ be the Riemannian connection on M . Substituting Y by φX in (1), we
obtain:

(∇Xφ)φX = (∇φXφ)X

for any X ∈ Γ(D) in which D = {X ∈ TM ; η(X) = 0}, or equivalently,

∇XX +∇φXφX = φ[φX,X].

Let ∇f−1TḾ denotes the pull-back of the Levi-Civita connection ∇′ on M´to the pull-
back bundle f−1TM´−→M . Since f is (φ, φ́)-holomorphic, we also have on M´:

∇f
−1TḾ
X df(X) +∇f

−1TḾ
φX df(φX) = φ́ [φ́ df(X), df(X)].



REFERENCES 953

If Bf is the second fundamental form of f , that means Bf satisfies Bf (X,Y ) =

∇f
−1TḾ
Y df(X)− df(∇XY ), it follows that for any X ∈ Γ(D), we have:

Bf (X,X) +Bf (φX, φX) = ∇f
−1TḾ
X df(X)− df(∇XX) +∇f

−1TḾ
φX df(φX)− df(∇φXφX).

= φ́ [φ́ df(X), df(X]− dfφ[φX,X]
= 0.

Let (e1, ..., en;φe1, ..., φen, ξ) be a local orthonormal φ-basis where {ei, φei} ∈ Γ(D),
then:

τ(f) = Σn
i=1Bf (ei, ei) + Σn

i=1Bf (φei, φei) +Bf (ξ, ξ) = Bf (ξ, ξ).

Since dfoφ = ±φ́odf , then φ́(df(ξ)) = ±df(φξ) = 0, implies that there exists a function k
on M such that dfx(ξx) = k(x)ξ́f(x) for any x ∈M. On the other hand, by Lemma 3.1(b),

we have respectively on M and M ,́ ∇ξξ = 0 and ∇ξ́ξ́ = 0, it follow that Bf (ξ, ξ) = 0 and
the proof is complete.

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