KYUNGPOOK Math. J. 2019; 59(4): 771-781
Biharmonic Submanifolds of Quaternionic Space Forms
Bouazza Kacimi and Ahmed Mohammed Cherif∗,1
Department of Mathematics, Faculty of Exact Sciences, Mascara University, Mascara, 29000, Algeria
e-mail : bouazza.kacimi@univ-mascara.dz and a.mohammedcherif@univ-mascara.dz
* Corresponding Author.
1 Laboratory of Geometry, Analysis, Controle and Applications, Algeria.
Received: August 1, 2018; Accepted: January 28, 2019; Published online: December 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we consider biharmonic submanifolds of a quaternionic space form. We give the necessary and sufficient conditions for a submanifold to be biharmonic in a quaternionic space form, we study different particular cases for which we obtain some non-existence results and curvature estimates.

Keywords: Quaternionic space form, biharmonic submanifold. This work was supported by National Agency Scientiﬁc Research of Algeria.
1. Introduction

Let (Mm, g) and (Nn, h) be two Riemannian manifolds. A harmonic map is a map ϕ : (Mm, g) → (Nn, h) that is a critical point of the energy functional

$E(ϕ)=12∫D∣dϕ∣2vM,$

for any compact domain D, where vM is the volume element [1, 6]. The Euler-Lagrange equation of E(ϕ) is

$τ(ϕ)=Tr(∇dϕ)=0,$

where τ (ϕ) is the tension field of ϕ [1, 6]. The map ϕ is said to be biharmonic if it is a critical point of the bienergy functional

$E2(ϕ)=12∫D∣τ(ϕ)∣2vM,$

for any compact domain D. In [11], Jiang obtained the Euler-Lagrange equation of E2(ϕ). This gives us

$τ2(ϕ)=Tr(∇ϕ∇ϕ-∇∇ϕ)τ(ϕ)-Tr(RN(dϕ,τ(ϕ))dϕ)=0,$

where τ2(ϕ) is the bitension field of ϕ and RN is the curvature tensor of N, ∇ϕ denote the pull-back connection on ϕ−1TN. Harmonic maps are always biharmonic maps by definition.

A submanifold in a Riemannian manifold is called a biharmonic submanifold if the isometric immersion defining the submanifold is a biharmonic map, and a biharmonic map is called a proper-biharmonic map if it is non-harmonic map. Also, we will call proper-biharmonic submanifolds a biharmonic submanifols which is non-harmonic [3, 7].

During the last decades, there are several results concerning the biharmonic submanifolds in space forms like real space forms [4], complex space forms [7], Sasakian space forms [8], generalized complex and Sasakian space forms [15], products of real space forms [14]. Motivated by this works, in this note, we will focus our attention on biharmonic submanifolds of quaternionic space form, we first give the necessary and sufficient condition for submanifolds to be biharmonic. Then, we apply this general result to many particular cases and obtain some non-existence results and curvature estimates.

2. Preliminaries

We recall some facts on quaternionic Kähler manifolds and their submanifolds. For a more detail we refer the reader, for example, to [2, 5, 10, 12, 13, 16].

An almost quaternionic structure on a smooth manifold N is a rank-three subbundle σEnd(TN) such that a local basis J = (Jα)α=1,2,3 exists of sections of σ satisfying

${Jα2=-IdJ1J2=-J2J1=J3$

where α = 1, 2, 3. The pair (N, σ) is called an almost quaternionic manifold.

Let (N, σ) be an almost quaternionic manifold. A metric tensor field g on N is called adapted to σ if the following compatibility condition holds

$g(JαX,JαY)=g(X,Y)$

for all local basis (Jα)α=1,2,3 of σ and X, Y ∈ Γ(TN). The triple (N, σ, g) is said to be an almost quaternionic Hermitian manifold. It is easy to see that any almost quaternionic Hermitian manifold has dimension 4n. An almost quaternionic Hermitian manifold (N, σ, g) is a quaternionic Kähler manifold if the Levi-Civita connection verifies

$∇Jα=ωγ⊗Jβ-ωβ⊗Jγ$

for any cyclic permutation (α, β, γ) of (1, 2, 3), (ωα)α=1,2,3 being local 1-forms over the open for which (Jα)α=1,2,3 is a local basis of σ.

A submanifold M of a quaternionic Kähler manifold N is called a quaternionic submanifold (resp. totally real submanifold) if Jα(TM) ⊂ TM (resp. Jα(TM) ⊂ TM), α = 1, 2, 3, where TM and TM denote the tangent and normal bundle of M, respectively.

Let (N, σ, g) be a quaternionic Kähler manifold and let X be a unit vector tangent to (N, σ, g). Then the 4-plane spanned by {X, J1X, J2X, J3X}, denoted by Q(X), is called a quaternionic 4-plane. Any 2-plane in Q(X) is called a quaternionic plane. The sectional curvature of a quaternionic plane is called a quaternionic sectional curvature. A quaternionic Kähler manifold is a quaternionic space form if its quaternionic sectional curvatures are equal to a constant, say 4c. We denote by N4n(4c) the quaternionic space form of constant quaternionic sectional curvature 4c. The Standard models of quaternionic space forms are the quaternionic projective space ℍPn(4c)(c > 0), the quaternionic space ℍn(c = 0) and the quaternionic hyperbolic space ℍHn(4c)(c < 0). The Riemannian curvature tensor R of a quaternionic space form N4n(4c) is of the form

$R(X,Y)Z=c{〈Z,Y〉X-〈X,Z〉Y+∑α=13[〈Z,JαY〉JαX-〈Z,JαX〉JαY+2〈X,JαY〉JαZ]}$

for X, Y, Z ∈ Γ(TN4n(4c)), where 〈·, ·〉 is the Riemannian metric on N4n(4c) and (Jα)α=1,2,3 is a local basis of σ.

Now, let M be a submanifold of a quaternionic space form N4n(4c). Then for any X ∈ Γ(TM), we write

$JαX=jαX+kαX$

where, jα : TMTM and kα : TMNM, here NM denote the normal bundle of M. Similarly, for any ξ ∈ Γ(NM), we have

$Jαξ=lαξ+mαξ$

where, lα : NMTM and kα : NMNM. Since for any α ∈ {1, 2, 3}, Jα satisfies (2.1) and (2.2). Then, we deduce that the operators jα, kα, lα, mα satisfy the following relations

$jα2X+lαkαX=-X,$$mα2ξ+kαlαξ=-ξ,$$jαlαξ+lαmαξ=0,$$kαjαX+mαkαX=0,$$g(kαX,ξ)=-g(X,lαξ),$$g(jαX,Y)=-g(X,jαY),$$g(mαξ,η)=-g(ξ,mαη).$

for all X, Y ∈ Γ(TM) and all ξ, η ∈ Γ(NM).

3. Biharmonic Submanifolds of Quaternonic Space Forms

Let Mm be a submanifold of N4n(4c), ϕ : MmN4n(4c) be the canonical inclusion. We shall denote by B, A, H, Δ and Δ the second fundamental form, the shape operator, the mean curvature vector field, the Laplacian and the Laplacian on the normal bundle of Mm in N4n(4c), respectively.

### Theorem 3.1

Let Mm be a submanifold of N4n(4c). Then Mm is biharmonic if and only if

${Δ⊥H+Tr(B(·,AH(·)))-mcH+3c∑α=13kαlαH=0,m2grad(∣H∣2)+2 Tr(A∇·⊥H·)+3c∑α=13jαlαH=0.$
Proof

Choose a local geodesic orthonormal frame {ei}1≤im at point p in Mm. Then calculating at p, by the use of the Gauss and Weingarten formulas, we have

$ΔH=-∑i=1m(∇eiϕ∇eiϕH)=-∑i=1m(∇eiϕ(-AHei+∇ei⊥H)) =-∑i=1m(-∇eiAHei-B(ei,AHei)-A∇ei⊥Hei+∇ei⊥∇ei⊥H) =Tr(∇.AH·)Tr B(·,AH·)+Tr(A∇·⊥H·)+Δ⊥H.$

Moreover,

$Tr(∇.AH·)=∑i=1m∇eiAH(ei)=∑i,j=1m∇ei(〈AH(ei),ej〉ej)=∑i,j=1m(ei〈AH(ei),ej〉)ej =∑i,j=1m(ei〈B(ej,ei),H〉)ej=∑i,j=1m(ei〈∇ejϕei,H〉)ej =∑i,j=1m(〈∇eiϕ∇ejϕei,H〉+〈∇ejϕei,∇eiϕH〉)ej =∑i,j=1m(〈∇eiϕ∇ejϕei,H〉+〈B(ej,ei),∇ei⊥H〉)ej =∑i,j=1m(〈∇eiϕ∇ejϕei,H〉+〈A∇ei⊥H(ei),ej〉)ej =∑i,j=1m〈∇eiϕ∇ejϕei,H〉ej+∑i=1mA∇ei⊥H(ei).$

Further, using (2.3) and (2.4) we have

$∑i,j=1m〈∇eiϕ∇ejϕei,H〉ej=∑i,j=1m〈RN4n(4c)(ei,ej)ei+∇ejϕ∇eiϕei+∇[ei,ej]ϕei,H〉ej =c∑i,j=1m∑α=13{〈〈ei,ej〉ei-〈ei,ei〉ej+〈ei,Jαej〉Jαei -〈ei,Jαei〉Jαej+2〈ei,Jαej〉Jαei,H〉ej} +∑i,j=1m〈∇ejϕB(ei,ei)+∇ejϕ∇eiei,H〉ej =3c∑i,j=1m∑α=13〈Jα(〈Jαej,ei〉ei),H〉ej+m∑j=1m〈∇ejϕH,H〉ej +∑i,j=1m〈∇ej∇eiei+B(ej,∇eiei),H〉ej =3c∑i,j=1m∑α=13〈jα2ej,H〉ej+m2∑j=1mej(∣H∣2)ej =3c∑i,j=1m∑α=13〈-ej-lαkαej,H〉ej+m2grad(∣H∣2) =m2grad(∣H∣2).$

Reporting (3.4) into (3.3), we find

$Tr(∇.AH·)=m2grad(∣H∣2)+∑i=1mA∇ei⊥H(ei).$

Replacing (3.5) into (3.2), we get the following formula

$ΔH=m2grad(∣H∣2)+Tr(B(·,AH(·)))+2 Tr(A∇·⊥H·)+Δ⊥H.$

Furthermore, we have

$τ(ϕ)=Tr(∇dϕ)=mH.$

From (1.1) and (3.7), we find

$τ2(ϕ)=Tr(∇ϕ∇ϕ-∇∇ϕ)τ(ϕ)-Tr(RN4n(4c)(dϕ,τ(ϕ))dϕ) =∑i=1m(∇eiϕ∇eiϕ-∇∇eieiϕ)mH-∑i=1mRN4n(4c)(dϕ(ei),mH)dϕ(ei) =-m{ΔH+∑i=1mRN4n(4c)(dϕ(ei),H)dϕ(ei)}$

By (2.3), we get

$∑i=1mRN4n(4c)(dϕ(ei),H)dϕ(ei)=-mcH+3c∑i=1m∑α=13〈lαH,ei〉Jαei =-mcH+3c∑α=13JαlαH =-mcH+3c∑α=13jαlαH+3c∑α=13kαlαH.$

Substituting (3.6) and (3.9) into (3.8), we obtain

$τ2(ϕ)=-m{m2grad(∣H∣2)+Tr(B(·,AH(·)))+2 Tr(A∇·⊥H·)+Δ⊥H-mcH +3c∑α=13jαlαH+3c∑α=13kαlαH}.$

Obviously jαlαH is tangent and kαlαH is normal for all α ∈ {1, 2, 3}, comparing the tangential and the normal parts, we get (3.1) and this completes the proof.

### Corollary 3.2

Let N4n(4c) be a quaternionic space form and M4n−1a real hypersurface of N4n(4c). Then M4n−1is biharmonic if and only if

${Δ⊥H+Tr(B(·,AH(·)))-4c(n+2)H=0,4n-12grad(∣H∣2)+2 Tr(A∇·⊥H·)=0.$
Proof

As M4n−1 is a hypersurface, then Jα for all α ∈ {1, 2, 3} maps normal vectors on tangent vectors, that is, mα = 0 for all α ∈ {1, 2, 3}. Hence, by relation (2.5), we have kαlαH = –H and by relation (2.6), jαlαH = 0, which gives the result by Theorem 3.1.

### Corollary 3.3

Let N4n(4c) be a quaternionic space form and Mm a totally real submanifold of N4n(4c). Then Mm is biharmonic if and only if

${Δ⊥H+Tr(B(·,AH(·)))-c(m+9)H=0,m2grad(∣H∣2)+2 Tr(A∇·⊥H·)=0.$
Proof

As Mm is a totally real submanifold, then jα = 0 for all α ∈ {1, 2, 3} and by the use of (2.6) we deduce that mα = 0 for all α ∈ {1, 2, 3}. Moreover, by relation (2.5), we have kαlαH = –H, which gives the proof by Theorem 3.1.

### Remark 3.4

In [5], Chen proved that every quaternionic submanifold of a quaternionic Kähler manifold is totally geodesic. Then we deduce that every quaternionic submanifold of N4n(4c) is biharmonic, and there exists no proper-biharmonic quaternionic submanifold in N4n(4c).

### Corollary 3.5

Let N4n(4c) be a quaternionic space form and Mm a totally real submanifold of N4n(4c) with parallel mean curvature vector field. Then it is biharmonic if and only if

$Tr(B(·,AH(·)))=c(m+9)H.$
Proof

Since Mm has parallel mean curvature, we obtain immediately the result from the Corollary 3.3.

### Proposition 3.6

Let M4n−1be a real hypersurface of N4n(4c) with non-zero constant mean curvature. Then M4n−1is proper-biharmonic if and only if

$∣B∣2=4c(n+2),$

or equivalently, M4n−1is proper-biharmonic if and only if the scalar curvature of M4n−1satisfies

$ScalM4n-1=c{(4n-1) (4n+7)-4n-17}+(4n-1)2∣H∣2.$
Proof

Assume thatM4n−1 is a real hypersurface of N4n(4c) with non-zero constant mean curvature. Then, by Corollary 3.2, the first equation of (3.11) becomes

$Tr(B(·,AH(·)))=4c(n+2)H.$

As, for hypersurfaces, we have AH = HA, then we can write

$Tr(B(·,AH(·)))=H Tr(B(·,A(·)))=H∣B∣2.$

Reporting (3.15) into (3.14), we get the identity (3.13).

For the second equivalence, by the use of the Gauss equation, we find

$ScalM4n-1=∑i,j=14n-1〈RN4n(4c)(ei,ej)ej,ei〉+∑i,j=14n-1〈B(ei,ei),B(ej,ej)〉-∑i,j=14n-1〈B(ej,ei),B(ej,ei)〉,$

where {ei}1≤i≤4n−1 is a local orthonormal frame of M4n−1. Therefore

$ScalM4n-1=∑i,j=14n-1〈RN4n(4c)(ei,ej)ej,ei〉+(4n-1)2∣H∣2-∣B∣2.$

Using (2.3) we have

$∑i,j=14n-1〈RN4n(4c)(ei,ej)ej,ei〉=c{(4n-1)2-(4n-1)+9(4n-1)-9} =c{(4n-1)(4n+7)-9}.$

Substituting (3.18) into (3.17), we get

$ScalM4n-1=c{(4n-1) (4n+7)-9}+(4n-1)2∣H∣2-∣B∣2.$

Hence, we deduce that M is proper-biharmonic if and only if |B|2 = 4c(n+2), that is, if and only if

$ScalM4n-1=c{(4n-1) (4n+7)-4n-17}+(4n-1)2∣H∣2.$

### Remark 3.7

The first equivalence of the previous proposition has been proven in [9] for the quaternionic projective space.

### Corollary 3.8

There exists no biharmonic real hypersurface with constant mean curvature in a quaternionic space form N4n(4c) of negative scalar curvature.

Proof

The assertion follows immediately from the first equivalence of Proposition 3.6.

### Example 3.9

The geodesic sphere S4n−1(u) of radius $u (0 in the quaternionic Euclidian space ℝ4n(= ℍn) is curvature adapted hypersurface of the quaternionic projective space ℍPn(4), i.e., Jαξ is a direction of the principal curvature for all α = 1, 2, 3, where ξ is the unit normal vector field along S4n−1 (see [2],[9]). Furthemore it is proper-biharmonic hypersurface in ℍPn(4) if and only if $(cot u)2=2n+7±2n2+4n+134n-1$. Indeed, the principal curvatures of S4n−1(u) are given as follows (see [2],[9]),

${λ1=cot u (with multiplicity m1=4(n-1)),λ2=2 cot(2u) (with multiplicity m2=3).$

The mean curvature H and the square of the second fundamental form |B|2 of S4n−1(u) are given by (see [9])

$H=14n-1{4(n-1)λ1+3λ2}, =14n-1{4(n-1) cot u+6 cot(2u)}, =4(n-1)4n-1t+34n-1(t-1t),$

where t = cot u.

$∣B∣2=4(n-1)λ12+3λ22 =4(n-1)t2+3(t-1t)2 =(4n-1)t2+3t2-6.$

On the other hand, using (3.19) we derive

$ScalS4n-1(u)=(4n-1) (4n+7)-9+(4n-1)2∣H∣2-∣B∣2.$

Then, by using the second equivalence of the Proposition 3.6, we have that S4n−1(u) is proper biharmonic if and only if

$(4n-1)t2+3t2-2(2n+7)=0⇔t2=2n+7±2n2+4n+134n-1,$

which has always solutions. We set for example $t=3$ for n = 2, then the geodesic sphere $S7(π6)$

is proper-biharmonic hypersurface in ℍP2(4).

In the next proposition we give an estimate of the mean curvature for a biharmonic totally real submanifold of ℍPn(4).

### Proposition 3.10

Let Mm a totally real submanifold ofPn(4) with non-zero constant mean curvature.

(1) If Mm is proper-biharmonic, then $0<∣H∣2≤m+9m$.

(2) If $∣H∣2=m+9m$, then Mm is proper-biharmonic if and only if it is pseudoumbilical andH = 0.

Proof

We assume that Mm is a biharmonic totally real submanifold of ℍPn(4) with non-zero constant mean curvature. By the first equation of (3.12), we have

$Δ⊥H+Tr(B(·,AH(·)))-(m+9)H=0.$

Then taking the scalar product of (3.21) with H, we find

$〈Δ⊥H,H〉+〈Tr(B(·,AH(·))),H〉-(m+9) 〈H,H〉=0.$

Since |H| is a constant, we get

$〈Δ⊥H,H〉=(m+9)∣H∣2-∣AH∣2.$

Using the Bochner formula, we have

$∣∇⊥H∣2+∣AH∣2=(m+9)∣H∣2.$

By using Cauchy-Schwarz inequality, i.e., |AH|2m|H|4 in the above equation, we obtain

$(m+9)∣H∣2≥m∣H∣4+∣∇⊥H∣2≥m∣H∣4.$

So, we can write

$0<∣H∣2≤m+9m,$

because |H| is a non-zero constant. This gives the proof of 1).

If$∣H∣2=m+9m$ and Mm is biharmonic. From (3.25), we derive ∇H = 0 and from (3.24), we have

$∣AH∣2=(m+9)2m.$

That is, Mm is pseudo-umbilical.

Conversely, if $∣H∣2=m+9m$ and Mm is pseudo-umbilical with ∇H = 0, then we get immediately

$Δ⊥H+Tr(B(·,AH(·)))-(m+9)H=0,$

and

$m2grad(∣H∣2)+2 Tr(A∇·⊥H·)=0.$

Therefore, by Corollary 3.3, Mm is proper-biharmonic. This completes the proof.

Acknowledgements

The authors would like to thank the reviewers for their useful remarks and suggestions.

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