SPECTRAL GEOMETRY OF HARMONIC MAPS INTO WARPED PRODUCT MANIFOLDS II

Let (Mn,g) be a closed Riemannian manifold and N a warped product manifold of two space forms. We investigate geometric properties by the spectra of the Jacobi operator of a harmonic map φ :M → N . In particular, we show if N is a warped product manifold of Euclidean space with a space form and φ,ψ : M → N are two projectively harmonic maps, then the energy of φ and ψ are equal up to constant if φ and ψ are isospectral. Besides, we recover and improve some results by Kang, Ki, and Pak (1997) and Urakawa (1989). 2000 Mathematics Subject Classification. 58C35, 58J10, 53C20.


Introduction.
In this paper, we deal with the inverse spectral problem of the Jacobi operator of a harmonic map from a compact manifold into warped product manifold.
The relationship between the geometry of a smooth manifold and the spectrum of the Laplacian has been studied by many authors (cf.[1,5,6]).In [6], Gilkey computed some spectral invariants concerning the asymptotic expansion of the trace of the heat kernel for an elliptic differential operator acting on the space of sections of a vector bundle (see also [5]).Urakawa applied the Gilkey's results to the Jacobi operator of a harmonic map from a closed (compact without boundary) manifold, M n , into a space form of constant curvature, N m (c), and proved that if the Jacobi operators of two harmonic maps from M into N have the same spectrum, then these harmonic maps have the same energy.The Jacobi operator of a harmonic map arises in the second variational formula of the energy functional and several people studied in this field (see [9,10,11,12]).In the case of Jacobi operator of a harmonic map, the spectral invariants computed by Gilkey can be expressed explicitly by the integration of geometric notions like curvature.
We will consider the Jacobi operator of a harmonic map from a closed manifold into a warped product manifold of two space forms which may be different.We generalize the results in [12] and prove some similar results about warped product manifolds.Warped product manifolds give us various examples and the structure of those are simple in some sense other than space forms (see [2]).Recently, Cheeger and Colding studied warped product manifolds and proved several remarkable results (see [4]).Also in [8], Ivanov and Petrova classified 4-dimensional Riemannian manifolds of positive constant curvature eigenvalues and showed that a warped product manifold is one of those manifolds and Gilkey, Leahy, and Sadofsky generalized this result for dimensions n = 5, 6, or n ≥ 9 (see [7]).

Preliminaries.
In this section, we describe, briefly, some results due to Gilkey and Urakawa about the asymptotic expansion of the trace of the heat kernel for the Jacobi operator of a harmonic map.
Let (M, g) be an n-dimensional compact Riemannian manifold without boundary and (N, h) an m-dimensional Riemannian manifold.A smooth map φ : M → N is said to be harmonic if it is a critical point of the energy functional E defined by where e(φ) = (1/2) h(φ * e i ,φ * e i ) called the energy density, φ * is the differential of φ, and {e i } is a local orthonormal frame of M. In other words, for any vector field where φ t : M → N is a one parameter family of smooth maps with φ 0 = φ and for every point x in M.
The second variational formula of the energy E for a harmonic map φ is given by Here J φ is a differential operator (called the Jacobi operator) acting on the space Γ (φ −1 T N) of sections of the induced bundle φ −1 T N. The operator J φ is of the form where ∇ is the connection of φ −1 T N which is induced by where V ∈ φ −1 T N, X is a tangent vector of M, ∇ h is the Levi-Civita connection of (N, h), and R N is the curvature tensor of (N, h).Since J φ is a selfadjoint, secondorder elliptic operator, and M is compact, J φ has a discrete spectrum of eigenvalues with finite multiplicities.We denote the spectrum of the Jacobi operator J φ of the harmonic map φ by The operator e −tJ φ is defined by where K(t, x, y, J φ ) is an endomorphism from the fiber of φ −1 T N at y to the fiber at x, called the kernel function.Then one has an asymptotic expansion for the L 2 -trace where a m (J φ ) is the spectral invariant of J φ which depends only on the spectrum, Spec(J φ ).Moreover, since M is compact and without boundary, the odd terms of a m vanish.For more detail, see [5,6].Finally, define the endomorphism L for φ −1 T N by (2.10) Then we have where Ric N denotes the Ricci curvature tensor of (N, h).Now applying Gilkey's results to the Jacobi operator of a harmonic map, one has the following theorem.
3. Spectral invariants for warped product manifolds.We now assume that the target manifold (N, h) is a warped product manifold of the form , where N m i (c i ) is a space form of constant curvature c i (i = 1, 2), and f is a positive smooth function defined on N m 1 (c 1 ).Furthermore, the Riemannian metric h is of the form h = h 1 +f 2 h 2 , where h i is the standard metric on N m i (c i ) with constant curvature c i .
We use the following convention for the Riemannian curvature tensor, and so denoting h = , we have in the space form of curvature c, and the vertical part , the energy density of φ splits as follows: Finally, we denote (3.6) Then we have where The rest of this section is devoted to compute the terms a 2 (J φ ) and a 4 (J φ ) of the asymptotic expansion for the Jacobi operator J φ in the case To compute them, we have to calculate the terms Tr N 2 , and Tr g (L 2 ) = L 2 .To do this, the following lemma is needed.From now on, M is a closed Riemannian manifold and unless otherwise stated.Lemma 3.1.Let X, Y , Z be vector fields on N m 1 (c 1 ) and U , V , W vector fields on N m 2 (c 2 ).Then the Riemannian curvature tensor R = R N of N satisfies the following: where D denotes the Riemannian connection on M, and ∇f denotes the gradient of f .
From now, we will compute norms of curvature tensors.

Tr g (L)
. Note that where m = m 1 + m 2 , and Using φ * e i = φ * e T i + φ * e ⊥ i , and Lemma 3.1, one can get On the other hand, since ∇f is a horizontal vector field, that is, the tangential where Ddf denotes the Hessian of f .Hence using these identities, one has (3.13) where φ * e ⊥ i ,φ * e ⊥ j φ * Ddf e i ,φ * Ddf e j . (3.16) Summing up these two equations, one gets (3.17)

Tr g (L 2
).Note that (3.18) A straightforward computation which is a little complicated, but not still hard shows In the last term one can use the following identity: Similarly one has Therefore, , where φ i = π i •φ (i = 1, 2), and π i : N → N m 1 (c i ) (i = 1, 2) be the projection.Then substituting (3.13), (3.17), and (3.22) into Theorem 2.1, one gets the following theorem.Theorem 3.2.Let φ : (M, g) → N m 1 (c 1 ) × f N m 2 (c 2 ) be a harmonic map of an n-dimensional compact Riemannian manifold (M, g) into an m(= m 1 +m 2 )-dimensional Riemannian warped product manifold N. Then the coefficients a 0 (J φ ), a 2 (J φ ), and a 4 (J φ ) of the asymptotic expansion for the Jacobi operator J φ are, respectively, given by where Note that the integration of the function f over M means the integration of f • φ 1 over M.
In the product case, that is, f is a constant function 1, Theorem 3.2 reduces to the following which is a result due to [9].However our expression looks a little more concrete.
be a harmonic map of an n-dimensional compact Riemannian manifold (M, g) into an m(= m 1 + m 2 )dimensional Riemannian product manifold N. Then the coefficients a 0 (J φ ), a 2 (J φ ), and a 4 (J φ ) of the asymptotic expansion for the Jacobi operator J φ are, respectively, given by is also harmonic.So Corollary 3.3 implies that the coefficients of the asymptotic expansion for the Jacobi operator J φ split as follows: (3.26) Also Corollary 3.3 reproves a result of [12].Corollary 3.5 (see [12]).Let φ : (M, g) → N m (c) be a harmonic map of an ndimensional compact Riemannian manifold (M, g) into an m-dimensional space form N. Then the coefficients a 0 (J φ ), a 2 (J φ ), and a 4 (J φ ) of the asymptotic expansion for the Jacobi operator J φ are, respectively, given by Corollary 3.6.Let φ : (M, g) → N m (c) × N m (c) be a harmonic map of an ndimensional compact Riemannian manifold (M, g) into a 2m-dimensional Riemannian product manifold N. Then the coefficients a 0 (J φ ), a 2 (J φ ), and a 4 (J φ ) of the asymptotic expansion for the Jacobi operator J φ are, respectively, given by M s g e(φ) dv g . (3.28)

Applications.
In this section, we will investigate properties for the Jacobi operator when the spectrum of the harmonic maps coincide in various cases of N.
First we recover a result of [9].
In Corollary 4.1, if furthermore M has a constant scalar curvature, then one has The following theorem is an improved version of [9,Corollary 3.3].and hence E(φ) = e(ψ).
In Corollary 4.2, if furthermore M has a constant scalar curvature, then one has Finally, we will discuss projectively harmonic maps.In general, the composition of two harmonic maps is not necessarily harmonic.Definition 4.3.We say a harmonic map φ : Not every harmonic map is in general projectively harmonic (cf.[3]).If ) is harmonic and so in this case every harmonic map φ : M → N is projectively harmonic.In particular, in the case of product (i.e., f ≡ 1), every harmonic map is automatically projectively harmonic.
be two projectively harmonic maps.If φ and ψ are isospectral, then where and A(ψ) is similar.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: One can consider φ as a map φ : M → N m (c)×N m (c) with φ = (φ, const.)and apply Corollary 3.3.As a special case of Corollary 3.3, when m 1 = m 2 = m and c 1 = c 2 = c, one gets the following corollary.