EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 150-159
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Biharmonic Maps into S-Space Forms

Najma Abdul Rehman1,∗, Mehwish Bari2

1 Department of Mathematics, CIIT, Sahiwal, Pakistan
2 Department of Mathematics, NCBA and E, Bahawalpur, Pakistan

Abstract. We study in this paper the condition on second fundamental form for biharmonicity
of submanifolds in S-space forms.

2010 Mathematics Subject Classifications: 53C55, 53C43, 58E20

Key Words and Phrases: S-Manifolds, harmonic map, biharmonic maps

1. Introduction

Harmonic maps between Riemannian manifolds have been discussed for last decades,
initiated with the paper of J. Eells and J.H. Sampson [7]. Since harmonic maps have both
properties analytic and geometric, they have become an important and interesting re-
search field. The study of harmonic maps on Riemannian manifolds with some structures
started from the paper of Lichnerowicz [13]. After that, Rawnsley [14] studied structure
preserving harmonic maps between f-manifolds. Later on many authors studied harmonic
maps (see [6] [10], [11], [12], [15] [16]).

The biharmonic maps theory is an old and attractive subject. They have been studied
since 1862 by Maxwell and Airy to describe a mathematical model of elasticity. The
Euler-Lagrange equation for bienergy functional was first derived by Jiange in 1986 [8].
After this biharmonic maps were studied by many authors see [2], [3], [5]. In [5], authors
have studied the biharmonic submanifolds in complex space form. The objective of this
paper is to find condition on second fundamental form for biharmonicity of a map from
submanifolds of S-space form to S-space forms. After we recall some well known facts
about biharmonic maps and S-manifolds , we prove the main results in third section.

∗Corresponding author.

Email addresses: najma−ar@hotmail.com (N. A. Rehman), mehwishbari@yahoo.com (M. Bari)

http://www.ejpam.com 150 c© 2018 EJPAM All rights reserved.



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 151

2. Preliminaries

In this section, we recall some well known facts concerning harmonic maps, biharmonic
maps and S-manifolds.
Let F : (M, g) −→ (N,h) be a smooth map between two Riemannian manifolds of dimen-
sions m and n respectively. The energy density of F is a smooth function e(F ) : M −→
[0,∞) given by [7],

e(F )p =
1

2
Trg(F

∗h)(p) =
1

2

m∑
i=1

h(F∗pui, F∗pui),

for any p ∈ M and any orthonormal basis {u1, . . . , um} of TpM . If M is a compact
Riemannian manifold, the energy E(F ) of F is the integral of its energy density:

E(F ) =

∫
M
e(F )υg ,

where υg is the volume measure associated with the metric g on M. A map F ∈ C∞(M,N)
is said to be harmonic if it is a cricital point of the energy functional E on the set of all
maps between (M, g) and (N, h). Now, let (M, g) be a compact Riemannian manifold.
If we look at the Euler-Lagrange equations for the corresponding variational problem, a
map F : M −→ N is harmonic if and only if τ(F ) ≡ 0, where τ(F ) is the tension field
which is defined by

τ(F ) = Trg∇̃dF,

where ∇̃ is the connection induced by the Levi-Civita connection on M and the F-pullback
connection of the Levi Civita connection on N.

We take now a smooth variation Fs,t with two parameters s, t ∈ (−ε, ε) such that
F0,0 = F . The corresponding variation vector fields are denoted by V and W.

The second variation formula of E is:

HF (V,W ) =
∂2

∂s∂t
(E(Fs,t))

∣∣
(s,t)=(0,0)

=

∫
M
h(JF (V ),W )υg,

where JF is a second order self-adjoint elliptic operator acting on the space of variation
vector fields along F (which can be identified with Γ(F−1(TN))) and is defined by

JF (V ) = −
m∑
i=1

(∇̃ui∇̃ui − ∇̃∇uiui
)V −

m∑
i=1

RN (V, dF (ui))dF (ui), (1)

for any V ∈ Γ(F−1(TN)) and any local orthonormal frame {u1, . . . , um} on M. Here RN

is the curvature tensor of (N, h) (see [9] for more details on harmonic maps).
J. Eells and L. Lemaire [9] proposed polyharmonic (k-harmonic) maps, and Jiang [8]



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 152

studied the first and second variation formulas of biharmonic maps. Let us consider the
bienergy functional defined by:

E2(F ) =
1

2

∫
M
| τ(F ) |2 νg, (2)

where | V |2= h(V, V ), V ∈ Γ(F−1TN).
Then, the first variation formula of the bienergy functional is given by:

d

dt
|t=0 E2(Ft) = −

∫
M
h(τ2(F ), V )νg, (3)

here

τ2(F ) := J(τ(F )) = 4̄(τ(F ))−R(τ(F )), (4)

which is called the bitension field of F and J is given by (1).
A smooth map F of (M, g) into (N, h) is said to be biharmonic if τ2(F ) = 0.

As a generalization of both almost complex (in even dimension) and almost contact (in
odd dimension) structures, Yano introduced in [17] the notion of f -structure on a smooth
manifold of dimension 2n + s, i.e. a tensor field of type (1,1) and rank 2n satisfying
f3 +f = 0. The existence of such a structure is equivalent to a reduction of the structural
group of the tangent bundle to U(n) × O(s). Let N be a (2n + s)-dimensional manifold
with an f -structure of rank 2n. If there exist s global vector fields ξ1, ξ2, . . . , ξs on N such
that:

fξα = 0, ηα ◦ f = 0, f2 = −I +
∑

ξα ⊗ ηα, (5)

where ηα are the dual 1-forms of ξα, we say that the f -structure has complemented frames.
For such a manifold there exists a Riemannian metric g such that

g(X,Y ) = g(fX, fY ) +
∑

ηα(X)ηα(Y )

for any vector fields X and Y on N . See [1].
An f -structure f is normal, if it has complemented frames and

[f, f ] + 2
∑

ξα ⊗ dηα = 0,

where [f, f ] is Nijenhuis torsion of f .
Let Ω be the fundamental 2-form defined by Ω(X,Y ) = g(X, fY ), X, Y ∈ T (N). A

normal f -structure for which the fundamental form Ω is closed, η1∧· · ·∧ηs∧(dηα)n 6= 0 for
any α, and dη1 = · · · = dηs = Ω is called to be an S-structure. A smooth manifold endowed
with an S-structure will be called an S-manifold. These manifolds were introduced by Blair
in [1].



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 153

We have to remark that if we take s = 1, S-manifolds are natural generalizations of
Sasakian manifolds. In the case s ≥ 2 some interesting examples are given in [1].

If N is an S-manifold, then the following formulas are true (see [1]):

∇Xξα = −fX, X ∈ T (N), α = 1, . . . , s, (6)

(∇Xf)Y =
∑
{g(fX, fY )ξα + ηα(Y )f2X}, X, Y ∈ T (N), (7)

where ∇ is the Riemannian connection of g. Let L be the distribution determined by the
projection tensor −f2 and let K be the complementary distribution which is determined
by f2 + I and spanned by ξ1, . . . , ξs. It is clear that if X ∈ L then ηα(X) = 0 for any α,
and if X ∈ K, then fX = 0. A plane section π on N is called an invariant f -section if
it is determined by a vector X ∈ L(x), x ∈ N, such that {X, fX} is an orthonormal pair
spanning the section. The sectional curvature of π is called the f -sectional curvature. If
N is an S-manifold of constant f -sectional curvature k, then its curvature tensor has the
form [1]

R(S, T, V,W ) =
∑
α,β

{g(fS, fW )ηα(T )ηβ(V )− g(fS, fV )ηα(T )ηβ(W ) +

+g(fT, fV )ηα(S)ηβ(W )− g(fT, fW )ηα(S)ηβ(V )}+

+
1

4
(k + 3s){g(fS, fW )g(fT, fV )− g(fS, fV )g(fT, fW )}+

+
1

4
(k − s){Ω(S,W )Ω(T, V )− Ω(S, V )Ω(T,W )− 2Ω(S, T )Ω(V,W )}, (8)

S, T, V, W ∈ T (N). Ω is fundamental 2-form. Such a manifold N(k) will be called an
S-space form. The Euclidean space E2n+s and the hyperbolic space H2n+s are examples
of S-space forms.

Let M be an m-dimensional submanifold immersed in N. Then M is an invariant subm-
naifold if ξα ∈ TM for any α and fV ∈ TM for any V ∈ TM . It is said to be anti-invariant
submanifold if fV ∈ TM⊥ for any V ∈ TM . For a vector field X ∈ TM⊥, it can be writ-
ten as fX = tX + nX, where tX is tangent component of fX, nX is normal component
of f X. If n does not vanishes, then its an f-structure [4].

Consider the structure vector fields ξ1, ξ2, . . . , ξs are tangent to M, dim(M) ≥ s. Then
M is CR-submaifold of N if there are two differentiable distributions D and D⊥ on M,
TM = D +D⊥ such that

• D and D⊥ are mutually orthogonal to each other.

• The distribution D is invariant under f, i.e. fDp = Dp, for any p ∈M

• The distribution D⊥ is anti invariant under f, i.e. fD⊥p ⊆ TpM⊥ for any p ∈M .

It can be proved that each hypersurface of N which is tangent to ξ1, ξ2, . . . , ξs, has the
structure of CR-submnaifold of N, for detail see [4].



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 154

3. Biharmonic Maps into S-space form

Before the main results recall the following results by Jiang:

Lemma 1. [8] Let f : (Mm, g)→ (Nn, h) be an isometric immersion whose mean curva-
ture vector field H = 1

mτ(f) is parallel; ∇⊥H = 0, where ∇⊥ is the induced connection of
the normal bundle T⊥M by f. Then,

4τ(f) =
m∑
i=1

h(4τ(f), df(ei))df(ei)−
m∑

i,j=1

h(∇̃eiτ(f), df(ej))(∇̃eidf)(ej),

where {ei} is a locally defined orthonormal frame field of (M, g).

Lemma 2. [8] Let f : (Mm, g)→ (Nn, h) be an isometric immersion whose mean curva-
ture vector field H = 1

mτ(f) is parallel; ∇⊥H = 0, where ∇⊥ is the induced connection of
the normal bundle T⊥M by f. Then,

4τ(f) = −
m∑

j,k=1

h(τ(f), RN (df(ej), df(ek))df(ej)df(ek)−

−
m∑

i,j=1

h(τ(f), (∇̃eidf)(ej))(∇̃eidf)(ej),

where {ei} is a locally defined orthonormal frame field of (M, g).

Lemma 3. [8] Let f : (Mm, g) → (Nm+1, h) be an isometric immersion which is not
harmonic. Then, the condition that ‖ τ(f) ‖ is constant is equivalent to the one that

∇Xτ(f) ∈ Γ(f∗TM), for all X ∈ TM,

that is the mean curvature tensor is parallel with respect to 5⊥.

For details and proof of these Lemmas, see [5], [8]. Now the main result of this article;

3.1. Main results

Theorem 1. Let (M, g) be a 2m+s-dimensional submanifold of S-space form N of dimen-
sion (2n+s), and Φ : (M, g) → (N,h) be an isometric immersion with non zero constant
parallel mean curvature with respect to connection on normal bundle, then necessary and
sufficient conditions for Φ to be biharmonic is

• ‖ B(Φ) ‖2= k+3s
4 (2n− 1 + s) + 3(k−s)

4 , for M2m+s to be a hypersurface, 2m=2n-1

• ‖ B(Φ) ‖2= k+3s
4 (2m+ s), for M2m+s to be an inavriant submanifold, m < n



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 155

Proof. Consider an s-manifold with constant f -sectional curvature k. Let {vi}2m+s
i=1 be

orthonormal basis on M . Then from equation (8) we have

RN (dΦ(vj), dΦ(vk))dΦ(vk) =
∑
α,β

{
−f2dΦ(vj)ηα(dΦvk)ηβ(dΦvk)− h(fdΦvj , dΦvk).

.ηα(dΦvk)ξβ + h(fdΦ(vk), fdΦ(vk))ηα(dΦvj)ξβ+

+f2dΦ(vk)ηα(dΦvj)ηβ(dΦvk)
}

+
1

4
(k + 3s)

{
−f2dΦ(vj).

.h(fdΦvk, fdΦvk) + h(fdΦvj , fdΦvk)f
2dΦ(vk)

}
+

1

4
(k − s) {−fdΦ(vj)h(dΦvk, dΦvk) + fdΦ(vk)

.h(dΦvj , fdΦvk) + 2fdΦ(vk)h(dΦvj , fdΦvk)} ,

and

RN (dΦ(vj), dΦ(vk))dΦ(vk) =
1

4
(k + 3s) {dΦ(vj) + δjk(−dΦvk)}

+
3

4
(k − s)h(dΦvj , fdΦvk)fdΦ(vk).

Then we have

m∑
j,k=1

h(τ(Φ), RN (dF (vj), dF (vk))dF (vk))dF (vj)

=
1

4
(k + 3s) {h(τ(Φ), dΦ(vj))− δjkh(τ(Φ), dΦvk)}

+
3

4
(k − s) {h(dΦvj , fdΦvk)h(τ, fdΦ(ek))} . (9)

Let τ(Φ) ∈ TM⊥, then

m∑
j,k=1

h
(
τ(Φ), RN (dΦ(vj), dΦ(vk)

)
dΦ(vk))dΦ(vj) = 0.

Let M2m+s be a hyperspace of s-manifold N. Each hypersurface of s-manifolds has the
structure of CR-submanifold. For τ(Φ) ∈ TM⊥, we can take fτ(Φ) ∈ ΓTM . Now
dim(M) = 2m + s = 2n − 1 + s. For orthonormal basis {dΦ(vk)}mk=1 of dΦ(TxM) at all
points on M, we can write

fτ(Φ) =
2n−1+s∑
k=1

h(fτ(Φ), dΦ(vk))dΦ(vk).

By Lemma 2,

∆τ(Φ) =

2n−1+s∑
i,j=1

h
(
τ(F ), (∇̃vidΦ)(vj)

)
(∇̃vidΦ)(vj)



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 156

Furthermore, we have

R(τ(Φ)) =

2n−1+s∑
k=1

RN (τ(Φ), dΦ(ek))dΦ =
k + 3s

4
(2n− 1 + s)τ(Φ) +

3(k − s)
4

τ(Φ). (10)

Now the necessary and sufficient conditions F to be biharmonic is that

τ2(F ) = 4̄τ(F )−R(τ(F )) = 0, (11)

this becomes

2n−1+s∑
i,j=1

h
(
τ(F ), ∇̃vjdF (vk)

)
∇̃vjdF (vk)−

[
k + 3s

4
(2n− 1 + s)τ(Φ) +

3(k − s)
4

τ(Φ)

]
= 0(12)

Now let

B(Φ)(ej , ek) = (∇̃vjdΦ)vk = h(dΦ(vj), dΦ(vk))V = hjkU,

where U is the unit normal vector along F (M). then

τ(F ) =

2n−1+s∑
r=1

(∇̃vrdF )(vr) =

2n−1+s∑
r=1

hrrU,

where U is the unit normal vector along F (M). Thus, the left hand side of (15) becomes
as:

2n−1+s∑
j,k,r=1

{
hrrhjkhjkU −

[
k + 3s

4
(2n− 1 + s)τ(Φ) +

3(k − s)
4

τ(Φ)

]}
= 0,

(
2n−1+s∑
r=1

hrr

)
m∑

j,k=1

hjkhjkU −
[
k + 3s

4
(2n− 1 + s)U +

3(k − s)
4

U

] = 0,

‖ τ(Φ) ‖
{
‖ B(Φ) ‖2 U −

[
k + 3s

4
(2n− 1 + s)U +

3(k − s)
4

U

]}
= 0.

since τ(Φ) 6= 0, by assumption, then we have

‖ B(Φ) ‖2 −
[
k + 3s

4
(2n− 1 + s) +

3(k − s)
4

]
= 0,

and

‖ B(Φ) ‖2=
[
k + 3s

4
(2n− 1 + s) +

3(k − s)
4

]
. (13)



N. A. Rehman, M. Bari / Eur. J. Pure Appl. Math, 11 (1) (2018), 150-159 157

Next let M2m+s be an invariant submanifold of s-manifold N. Then for τ(Φ) ∈ ΓTM⊥,
fτ(Φ) ∈ ΓTM⊥. For orthonormal basis {dΦ(vk)}mk=1 of dΦ(TxM) at all points x ∈M , we
have h(fτ(Φ), dΦ(vk)) = 0, then in this case

R(τ(Φ)) =

2n−1+s∑
k=1

RN (τ(Φ), dΦ(ek))dΦ =
k + 3s

4
(2m+ s)τ(Φ). (14)

Then by equation (11),

2m+s∑
i,j=1

h
(
τ(F ), ∇̃vjdF (vk)

)
∇̃vjdF (vk)−

k + 3s

4
(2m+ s)τ(Φ) = 0 (15)

With similar computations as above we have

‖ τ(Φ) ‖
{
‖ B(Φ) ‖2 U −

[
k + 3s

4
(2m+ s)U

]}
= 0.

This implies

‖ B(Φ) ‖2= k + 3s

4
(2m+ s)

Corollary 1. Let (M, g) be a 2n−1-dimensional submanifold of complex space form N of
dimension 2n, and Φ : (M, g)→ (N,h) be an isometric immersion with non zero constant
parallel mean curvature with respect to connection on normal bundle, then necessary and
sufficient conditions for Φ to be biharmonic is

‖ B(Φ) ‖2= k

2
(n+ 1)

Proof. In equation (13), for s=0 we get result.

Corollary 2. Let (M, g) be a (2n − 1) + 1-dimensional submanifold of Sasakian space
form N of dimension 2n+1, and Φ : (M, g)→ (N,h) be an isometric immersion with non
zero constant parallel mean curvature with respect to connection on normal bundle, then
necessary and sufficient conditions for Φ to be biharmonic is

‖ B(Φ) ‖2= k

4
(2n+ 3) +

3

4
(2n− 1)

Proof. In equation (13), for s=1 we get result.

Acknowledgements

First Author is supported by NRPU research project of HEC , Pakistan.



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