k-harmonic maps into a Riemannian manifold with constant sectional curvature

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.


Introduction
Theory of harmonic maps has been applied into various fields in differential geometry. The harmonic maps between two Riemannian manifolds are critical maps of the energy functional E(φ) = 1 2 M dφ 2 v g , for smooth maps φ : M → N . On the other hand, in 1981, J. Eells and L. Lemaire [6] proposed the problem to consider the k-harmonic maps: they are critical maps of the functional E k (φ) = M e k (φ)v g , (k = 1, 2, · · · ), where e k (φ) = 1 2 (d + d * ) k φ 2 for smooth maps φ : M → N . G.Y. Jiang [3] studied the first and second variational formulas of the bi-energy E 2 , and critical maps of E 2 are called biharmonic maps (2-harmonic maps). There have been extensive studies on biharmonic maps.
In 1989, Wang Shaobo [9] studied the first variational formula of the k-energy E k , whose critical maps are called k-harmonic maps. Harmonic maps are always kharmonic maps by definition. But, the author [7] showed biharmonic is not always k-harmonic (k ≥ 3). More generally, s-harmonic is not always k-harmonic (s < k). Furthermore, the author [7] showed the second variational formula of the k-energy.
In this paper, we study k-harmonic maps into a Riemannian manifold with constant sectional curvature K.
In §1, we introduce notation and fundamental formulas of the tension field. In §2, we recall k-harmonic maps.
In §3, we give the relationship between biharmonic maps and k-harmonic maps.
In §4, we study 3-harmonic maps into a non positive sectional curvature and obtain non-existence theorem.
Finally, in §5, we define k-harmonic submanifolds of Euclidean spaces. And we show k-harmonic curve is a straight line. Furthermore, we give a conjecture for k-harmonic submanifolds in Euclidean spaces.

Preliminaries
Let (M, g) be an m dimensional Riemannian manifold, (N, h) an n dimensional one, and φ : M → N , a smooth map. We use the following notation. The second Here, ∇, ∇ N , ∇, ∇ are the induced connections on the bundles T M , If M is compact, we consider critical maps of the energy functional where e(φ) = 1 2 dφ 2 = m i=1 1 2 dφ(e i ), dφ(e i ) which is called the enegy density of φ, the inner product ·, · is a Riemannian metric h, and {e i } m i=1 is a locally defined orthonormal frame field on (M, g). The tension field τ (φ) of φ is defined by The curvature tensor field R N (·, ·) of the Riemannian metric on the bundle T N is defined as follows : , is the rough Laplacian. And G.Y.Jiang [3] showed that φ : (M, g) → (N, h) is a biharmonic (2-harmonic) if and only if △τ (φ) − R N (τ (φ), dφ(e i ))dφ(e i ) = 0.
2. k-harmonic maps J. Eells and L. Lemaire [6] proposed the notation of k-harmonic maps. The Euler-Lagrange equation for the k-harmonic maps was shown by Wang Shaobo [9]. In this section, we recall k-harmonic maps.
We consider a smooth variation {φ t } t∈Iǫ (I ǫ = (−ǫ, ǫ)) of φ with parameter t, i.e., we consider the smooth map F given by The corresponding variational vector field V is given by

k-HARMONIC MAPS INTO A RIEMANNIAN MANIFOLD WITH CONSTANT SECTIONAL CURVATURE 3
Definition 2.1 ( [6]). For k = 1, 2, · · · the k-energy functional is defined by We say for a k-harmonic map to be proper if it is not harmonic.
is a locally defined orthonormal frame field on (M, g). where, where, is a locally defined orthonormal frame field on (M, g).

The relationship between biharmonic and k-harmonic
In [7], the auther showed s-harmonic is not always k-harmonic (s < k). Especially, biharmonic is not always k-harmonic (k ≥ 3). So we study the relationship between biharmonic and k-harmonic (2 < k). We obtain some results. Proof. φ is biharmonic if and only if Thus, we have the proposition.
Proof. By using Proposition 3.1, we have Proof. By using Proposition 3.1, we have where, in the last equation, we only notice that Using these lammas, we show the following two theorems.

Proof. By Theorem 2.2, φ is 2s-harmonic if and only if
By Proposition 3.1, Lemma 3.2 and 3.3, we have Thus, we have the theorem.

Proof. By Theorem 2.3, φ is (2s + 1)-harmonic if and only if
By Proposition 3.1, Lemma 3.2 and 3.3, we have Thus, we have the theorem.

3-harmonic maps into non-positive curvature
In this section we show non-existence theorem of 3-harmonic maps. G. Y. Jiang showed the follows. We consider this theorem for 3-harmonic maps. First, we recall following theorem.
Using this theorem, we obtain the next result. Proof. Indeed, by computing the Laplacian of the 4-energy density e 4 (φ), we have due to φ is 3-harmonic. Here, we consider the right hand side of (7), Using Green's theorem, we have Then, the both terms of (8) are non-negative, so we have Especially, we have ∇ ei △τ (φ) = 0.
Using Theorem 4.2, we obtain the proposition.

k-harmonic curves into Euclidean space
In this section, we consider k-harmonic curves into a Euclidean space E n and we give a conjecture. B. Y. Chen [1] define biharmonic submanifolds of Euclidean spaces. B. Y. Chen and S. Ishikawa [2] proved that any biharmonic surface in E 3 is minimal. And Chen [1] gave a conjecture . There are several results for this conjecture ( [8], [5] and [10] etc). However, the conjecture is still open. I. Dimitric [5] considered a cureve case (n = 1), and obtained following theorem.

Theorem 5.3 ([5])
. Let x : C → E n be a smooth curve parametrized by arc length, with the mean curvature vector H satisfying △H = 0, then the curve is a straight line, i.e., totally geodesic in E n .
We generalize this throrem. First, we define k-harmonic submanifolds in Euclidean spaces.
Definition 5.4. Let x : M → E n be an isometric immersion into a Euclidean space.
x : M → E n is called k-harmonic submanifold if where, H = − 1 m △ x is the mean curvature vetor of the isometric immersion x and △ the Laplacian of M .
We also consider a curve case (n = 1), and obtain following theorem.
Theorem 5.5. Let x : C → E n be a smooth curve parametrized by arc length, with the mean curvature vector H satisfying △ k−1 H = 0, (k = 1, 2, · · · ) , then the curve is a straight line, i.e., totally geodesic in E n .
In other words, x(s) = a 1 s + a 0 with |a 1 | 2 = 1, and therefore the curve is a straight line.
Conjecture 5.6. The only k-harmonic submanifolds in Euclidean spaces are the minimal ones.
Especially, when k = 2, it is B. Y. Chen conjecture.