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

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

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

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

where τ₂(ϕ) is the bitension field of ϕ and R^N 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.

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 σ ⊂ End(TN) such that a local basis J = (J_α)_α=1,2,3 exists of sections of σ satisfying

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

for all local basis (J_α)_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

for any cyclic permutation (α, β, γ) of (1, 2, 3), (ω_α)_α=1,2,3 being local 1-forms over the open for which (J_α)_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 J_α(TM) ⊂ TM (resp. J_α(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

for X, Y, Z ∈ Γ(TN⁴ⁿ(4c)), where 〈·, ·〉 is the Riemannian metric on N⁴ⁿ(4c) and (J_α)_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

where, j_α : TM → TM and k_α : TM → NM, here NM denote the normal bundle of M. Similarly, for any ξ ∈ Γ(NM), we have

where, l_α : NM → TM and k_α : NM → NM. Since for any α ∈ {1, 2, 3}, J_α satisfies (2.1) and (2.2). Then, we deduce that the operators j_α, k_α, l_α, m_α satisfy the following relations

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

Let M^m be a submanifold of N⁴ⁿ(4c), ϕ : M^m → N⁴ⁿ(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 M^m in N⁴ⁿ(4c), respectively.

Theorem 3.1

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

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

Corollary 3.3

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

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 M^m a totally real submanifold of N⁴ⁿ(4c) with parallel mean curvature vector field. Then it is biharmonic if and only if

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

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

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.

Example 3.9

The geodesic sphere S⁴ⁿ⁻¹(u) of radius u (0<u<π2) in the quaternionic Euclidian space ℝ⁴ⁿ(= ℍⁿ) is curvature adapted hypersurface of the quaternionic projective space ℍPⁿ(4), i.e., J_αξ 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]),

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

where t = cot u.

On the other hand, using (3.19) we derive

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

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 M^m a totally real submanifold of ℍPⁿ(4) with non-zero constant mean curvature.

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

(2) If ∣H∣2=m+9m, then M^m is proper-biharmonic if and only if it is pseudoumbilical and ∇^⊥H = 0.

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