Article
KYUNGPOOK Math. J. 2019; 59(4): 771-781
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
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
Received: August 1, 2018; Accepted: January 28, 2019
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 Scientifi,c Research of Algeria.
1. Introduction
Let (
for any compact domain
where
for any compact domain
where
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
where
Let (
for all local basis (
for any cyclic permutation (
A submanifold
Let (
for
Now, let
where,
where,
for all
3. Biharmonic Submanifolds of Quaternonic Space Forms
Let
Theorem 3.1
Choose a local geodesic orthonormal frame {
Moreover,
Further, using (
Reporting (
Replacing (
Furthermore, we have
From (
By (
Substituting (
Obviously
Corollary 3.2
As
Corollary 3.3
As
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
Corollary 3.5
Since
Proposition 3.6
Assume that
As, for hypersurfaces, we have
Reporting (
For the second equivalence, by the use of the Gauss equation, we find
where {
Using (
Substituting (
Hence, we deduce that
Remark 3.7
The first equivalence of the previous proposition has been proven in [9] for the quaternionic projective space.
Corollary 3.8
The assertion follows immediately from the first equivalence of Proposition 3.6.
Example 3.9
The geodesic sphere
The mean curvature
where
On the other hand, using (
Then, by using the second equivalence of the Proposition 3.6, we have that
which has always solutions. We set for example
is proper-biharmonic hypersurface in ℍ
In the next proposition we give an estimate of the mean curvature for a biharmonic totally real submanifold of ℍ
Proposition 3.10
(1)
(2)
We assume that
Then taking the scalar product of (
Since |
Using the Bochner formula, we have
By using Cauchy-Schwarz inequality, i.e., |
So, we can write
because |
If
That is,
Conversely, if
and
Therefore, by Corollary 3.3,
Acknowledgements
The authors would like to thank the reviewers for their useful remarks and suggestions.
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