#### Abstract

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if is a biwave map into a Riemannian manifold under certain circumstance, then is a wave map. We verify that if is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then is a wave map. We finally obtain a theorem involving an unstable biwave map.

#### 1. Introduction

Harmonic maps between Riemannian manifolds were first introduced and established by Eells and Sampson [1] in 1964. Afterwards, there were two reports on harmonic maps by Eells and Lemaire [2, 3] in 1978 and 1988. Biharmonic maps, which generalized harmonic maps, were first studied by Jiang [4, 5] in 1986. In this decade, there has been progress in biharmonic maps made by Caddeo et al. [6, 7], Loubeau and Oniciuc [8], Montaldo and Oniciuc [9], Chiang and Wolak [10], Chiang and Sun [11, 12], Chang et al. [13], Wang [14, 15], and so forth.

Wave maps are harmonic maps on Minkowski spaces, and their equations are the second-order hyperbolic systems of partial differential equations, which are related to Einsteinβs equations and Yang-Mills fields. In recent years, there have been many new developments involving local well-posedness and global-well posedness of wave maps into Riemannian manifolds achieved by Klainerman and Machedon [16, 17], Shatah and Struwe [18, 19], Tao [20, 21], Tataru [22, 23], and so forth. Furthermore, Nahmod et al. [24] also studied wave maps from into (compact) Lie groups or Riemannian symmetric spaces, that is, gauged wave maps when , and established global existence and uniqueness, provided that the initial data are small. Moreover, Chiang and Yang [25] , Chiang and Wolak [26] have investigated exponential wave maps and transversal wave maps.

Bi-Yang-Mills fields, which generalize Yang-Mills fields, have been introduced by Ichiyama et al. [27] recently. The following connection between bi-Yang Mills fields and biwave equations motivates one to study biwave maps.

Let be a principal fiber bundle over a manifold with structure group and canonical projection , and let be the Lie algebra of . A connection can be considered as a -valued 1-form locally. The curvature of the connection is given by the 2-form with The bi-Yang-Mills Lagrangian is defined where is the adjoint operator of the exterior differentiation on the space of -valued smooth forms on (, the endormorphisms of ). Then the Euler-Lagrange equation describing the critical point of (1.2) has the form which is the bi-Yang-Mills system. In particular, letting and , the group of orthogonal transformations on we have that is a skew symmetric matrix The appropriate equivariant ansatz has the form where is a spatially radial function. Setting and the bi-Yang-Mills system (1.3) becomes the following equation for : which is a linear nonhomogeneous biwave equation, where is a function of and .

Biwave maps are biharmonic maps on Minkowski spaces. It is interesting to study biwave maps since their equations are the fourth-order hyperbolic systems of partial differential equations, which generalize wave maps. This is the first attempt to study biwave maps and their relationship with wave maps. There are interesting and difficult problems involving local well posedness and global well posedness of biwave maps into Riemannian manifolds or Lie groups (or Riemannian symmetric spaces), that is, gauged biwave maps for future exploration.

In Section 2, we compute the first variation of the bi-energy functional of a biharmonic map using tensor technique, which is different but much easier than Jiangβs [4] original computation. In Section 3, we prove in Theorem 3.3 that * if *ββ*is a biwave map and *ββ* is a totally geodesic map, then *ββ* is a biwave map. * Then we can apply this theorem to provide many biwave maps (see Example 3.4). We also can construct biwave nonwave maps as follow: * Let *ββ*be a wave map on a compact domain and let *ββ* be an inclusion map. The map *ββ* is a biwave nonwave map if and only if *ββ* has constant energy density, * compare with Theorem 3.5. Afterwards, we show that * if *ββ*is a biwave map on a compact domain into a Riemannian manifold satisfying *
then is a wave map (cf. Theorem 3.6). This theorem is different than the theorem obtained by Jiang [4]: * if *ββ* is a biharmonic map from a compact manifold into a Riemannian manifold with nonpositive curvature, then *ββ* is a harmonic map. * In Section 4, we verify that * if *ββ* is a stable biwave map into a ** Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then *ββ* is a wave map * (cf. Theorem 4.5). We also prove that * if *ββ* is a wave map on a compact domain with constant energy density, then *ββ*is an unstable biwave map * (cf. Theorem 4.7).

#### 2. Biharmonic Maps

A biharmonic map from an -dimensional Riemannian manifold into an -dimensional Riemannian manifold is the critical point of the bi-energy functional where is the volume form on .

*Notations*

is the adjoint of and is the tension field. Here is the Riemannian connection on induced by the Levi-Civita connections on and , and is the local frame at a point of . The tension field has components
where and are the Christoffel symbols on and , respectively.

In order to compute the Euler-Lagrange equation of the bi-energy functional, we consider a one-parameter family of maps from a compact manifold (without boundary) into a Riemannian manifold . Here is the endpoint of a segment starting at (=), determined in length and direction by the vector field along For a nonclosed manifold , we assume that the compact support of is contained in the interior of (we need this assumption when we compute by applying the divergence theorem). Then we have

Let The components of are We can use the curvature formula on and get where is the Riemannian curvature of . But therefore, has components We can rewrite (2.3) as where is a linear equation for and is an operator from to Solutions of are called Jacobi fields. Hence, we obtain the following definition from (2.3), (2.5), and (2.6).

*Definition 2.1. * is a biharmonic map if and only if the bitension field
that is, the tension field , is a Jacobi field.

If , then . Thus, harmonic maps are obviously biharmonic. Biharmonic maps satisfy the fourth-order elliptic systems of PDEs, which generalize harmonic maps. Our computation for the first variation of the bi-energy functional presented here using tensor technique is different but much easier than Jiangβs [4] original computation (it took him four pages).

Caddeo et al. [7] showed that a biharmonic curve on a surface of nonpositive Gaussian curvature is a geodesic (i.e., is harmonic) and gave examples of biharmonic nonharmonic curves on spheres, ellipses, unduloids, and nodoids.

Theorem 2.2 (see [4]). *Let be an isometric embedding of an -dimensional compact Riemannian manifold into an -dimensional unit sphere with nonzero constant mean curvature. The map is biharmonic if and only if where is the second fundamental form of *

*Example 2.3. *In the compact hypersurfaces, whose Gauss maps are isometric embeddings, are the Clifford surfaces [28]:
Let be a standard embedding such that Because and is a biharmonic nonharmonic map by Theorem 2.2.

#### 3. Biwave Maps

Let be an dimensional Minkowski space with the metric and the coordinates and let be an -dimensional Riemannian manifold. A wave map is a harmonic map on the Minkowski space with the energy functional The Euler-Lagrange equation describing the critical point of (3.1) is where is the wave operator on and are the Christoffel symbols of . is a wave map iff the wave field (i.e., the tension field on a Minkowski space) vanishes. The wave map equation is invariant with respect to the dimensionless scaling But, the energy is scale invariant in dimension .

If is a smooth map from a Minkowski space into a Riemannian manifold , then the bi-energy functional is, from (2.1), The Euler-Lagrange equation describing the critical point of (3.3), from (2.5), is

*Definition 3.1. * from a Minkowski space into a Riemannian manifold is a biwave map if and only if the biwave field (i.e., the bitension field on a Minkowski space),
that is, the wave field , is a Jacobi field on the Minkowski space.

Biwave maps satisfy the fourth-order hyperbolic systems of PDEs, which generalize wave maps. If , then . Waves maps are obviously biwave maps, but biwave maps are not necessarily wave maps.

*Example 3.2. *Let be a function defined on a Minkowski space satisfying the following conditions:
where the initial data and are given. Since this is a fourth-order homogeneous linear biwave equation with constant coefficients, it is well known that can be solved by [18, 29].

Let be a smooth map from a Minkowski space into a Riemannian manifold and let be a smooth map between two Riemannian manifolds and . Then the composition is a smooth map. Since is a semi-Riemannian manifold (i.e., a pseudo-Riemannian manifold), we can define a Levi-Civita connection on by OβNeill [30]. Let , be the connections on , respectively, and let be the curvatures on , respectively. We first have the following two formulas: for and for

Theorem 3.3. *If is a biwave map and is totally geodesic between two Riemannian manifolds and then the composition is a biwave map.*

*Proof. *Let be the coordinates of a point in and let be the frame at . We know from [4] that Since is totally geodesic, we have by applying the chain rule of the wave field to as [1]. Then we get
Recalling that we derive from (3.7a) that
since is totally geodesic. Therefore, we have
Substituting (3.10) into (3.8), we arrive at
where

On the other hand, we have by (3.7b)
We obtain from (3.11) and (3.12)
that is, Hence, if is a biwave map and is totally geodesic, then is a biwave map. Note that the total geodesicity of cannot be weakened into a harmonic or biharmonic map.

*Example 3.4. *Let be a submanifold of . Are the biwave maps into also biwave maps into ? The answer is affirmative iff is a totally geodesic submanifold of , that is, geodesics are geodesics. is a geodesic with iff is parallel, that is, iff For a map letting we have by (3.13) the following:
since is a geodesic. Hence, is a biwave map if and only if solves the fourth-order homogeneous linear biwave equation as in Example 3.2. It follows from Theorem 3.3 that there are many biwave maps provided by geodesics of .

We also can construct examples of biwave nonwave maps from some wave maps with constant energy using Theorem 3.5. Let be a hypersphere of Then is a biharmonic nonminimal submanifold of by Theorem 2.2 and Example 2.3. Let be a unit section of the normal bundle of in Then the second fundamental form of the inclusion is By computation, the tension field of is , and the bitension field is .

Theorem 3.5. *Let be a nonconstant wave map on a compact space-time domain and let be an inclusion. The map is a biwave nonwave map if and only if has constant energy density .*

*Proof. *Let be the coordinate of a point in and let be the frame at . Recall that is the unit section of the normal bundle. By applying the chain rule of the wave field to we have
since is a wave map. We can derive the following at the point by straightforward calculation:
Therefore, we obtain
Suppose that is a biwave nonwave map (). As the -part of , vanishes, which implies that is constant since is compact. The converse is obvious.

Let be the coordinates of a point in a compact space-time domain and be the frame at the point. Suppose that is a biwave map from a compact domain into a Riemannian manifold such that the compact supports of and are contained in the interior of

Theorem 3.6. *If is a biwave map from a compact domain into a Riemannian manifold such that
**
then is a wave map.*

*Proof. *Since is a biwave map, we have by (3.4)
Recall that are the coordinates of a point in and We compute
By applying the Bochnerβs technique from (3.19) and the assumption that the compact supports of and are contained in the interior of we know that is constant, that is, . If we use the identity
and the fact then we can conclude that by applying the divergence theorem.

Corollary 3.7. *If is a biwave map on a compact domain such that and , then is a wave map.*

*Proof. *The result follows from (3.19) immediately.

#### 4. Stability of Biwave Maps

Let be the coordinates of a point in a compact space-time domain and let be the frame at the point. Suppose that is a biwave map from a compact space-time domain into a Riemannian manifold such that the compact supports of and are contained in the interior of Let be a vector field such that . If we apply the second variation of a biharmonic map in [4] to a biwave map, we can have the following.

Lemma 4.1. *If is a biwave map from a compact domain into a Riemannian manifold, then
**
where is the Riemannian connection on and is the vector field along *

*Definition 4.2. *Let be a biwave map. If then is a stable biwave map.

If we consider a wave map, that is, as a biwave map, then by (4.1) we have and is automatically stable.

*Definition 4.3. *Let be a smooth map from a Minkowski space into a Riemannian manifold The *stress energy * is defined by where is the energy function and The map satisfies the *conservation *ββ*law * if

Proposition 4.4. *Let be a smooth map from a Minkowski space into a Riemannian manifold Then
*

*Proof. *Let be the coordinates of a point in and where is an matrix. We calculate
where the first term and the third term are canceled out and

Theorem 4.5. *Let be a compact domain and let be a Riemannian manifold with constant sectional curvature If is a stable biwave map satisfying the conservation law, then is a wave map.*

*Proof. *Because has constant sectional curvature, the second term of (4.1) disappears and (4.1) becomes
In particular, let Recalling that is a biwave map and has constant sectional curvature , (4.4) can be reduced to
Since satisfies the conservation law, by Definition 4.3, Proposition 4.4, and (4.2) we have
Substituting (4.6) into (4.5) and applying the stability of , we get
which implies that that is, is a wave map.

If we apply the Hessian of the bi-energy of a biharmonic map [4] to a biwave map , then we have the following.

Lemma 4.6. *Let be a biwave map. The Hessian of the bi-energy functional of is
**
where
**
for *

Theorem 4.7. *Let be a wave map on a compact domain with constant energy and let be an inclusion map. Then is an unstable biwave map. *

*Proof. *We have the following identities from Theorem 3.5:
Then we obtain the following formula from Lemma 4.6 and the previous identities:
which is strictly negative, where . Hence, is an unstable biwave map.

#### Acknowledgment

The author would like to appreciate Professor Jie Xiao and the referees for their comments.