On a Riemannian geometry, harmonic maps play a central role to study the geometric properties. They are critical points of the energy functional E(ϕ) for smooth maps ϕ:(M,g)→(M′,g′), where

E(ϕ)=12∫M|dϕ|2μM,

where μM is the volume element. It is well known that harmonic map is a solution of the Euler-Largrange equation τ(ϕ)=0, where τ(ϕ)=trg(∇dϕ) is the tension field.

In 1983, J. Eells and L. Lemaire extended the notion of harmonic map to biharmonic map, which is a critical points of the bienergy functional E2(ϕ), where

E2(ϕ)=12∫M|τ(ϕ)|2μM.

It is well-known [9] that harmonic maps are always biharmonic. But the converse is not true. At first, B.Y. Chen [3] raised so called Chen's conjecture and later, R. Caddeo et al. [2] raised the generalized Chen's conjecture. That is,

Generalized Chen's conjecture: Every biharmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic.

About the generalized Chen's conjecture, Nakauchi et al. [19] showed the following.

Theorem 1.1. [19] Let (M,g) be a complete Riemannian manifold and (M',g') be of non-positive sectional curvature. Then

• (1) every biharmonic map ϕ:M→M′ with finite energy and finite bienergy must be harmonic.

• (2) In the case Vol(M)=∞, every biharmonic map with finite bienergy is harmonic.

Now, we study the generalized Chen's conjecture for biharmonic maps on foliated Riemannian manifolds and extend Theorem 1.1 to foliations. Let (M,g,F) and (M′,g′,F′) be the foliated Riemannian manifolds. Let ϕ:M→M′ be a smooth foliated map, that is, map preserving the leaves. Then ϕ is said to be (F,F′)-harmonic map [6] if ϕ is a critical point of the transversal energy EB(ϕ), which is given by

EB(ϕ)=12∫M|dTϕ|2μM,

where dTϕ=dϕ|Q is the differential map of ϕ restricted to the normal bundle Q of F. From the first variational formula for the transversal energy functional [12], it is trivial that (F,F′)-harmonic map is a solution of τ˜b(ϕ):=τb(ϕ)+dTϕ(κB)=0, where τb(ϕ)=trQ(∇trdTϕ) is the transversal tension field and κB is the basic part of the mean curvature form κ of F.

The map ϕ is said to be (F,F′)-biharmonic map if ϕ is a critical point of the transversal bienergy functional E˜B,2(ϕ), where

E˜B,2(ϕ)=12∫M|τ˜b(ϕ)|2μM.

By the first variation formula for the transversal bienergy functional E˜B,2(ϕ) (Theorem 3.7), we know that (F,F′)-harmonic map is always (F,F′)-biharmonic. But the converse is not true. So we prove the generalized Chen's conjecture for (F,F′)-biharmonic map. That is, we prove the following theorem

Theorem 1.2. (cf. Theorem 3.10) Let (M,g,F) be a foliated Riemannian manifold and let (M′,g′,F′) be of non-positive transversal sectional curvature K Q′, that is, K Q′≤0. Let ϕ:M→M′ be a (F,F′)-biharmonic map. Then

• (1) if M is closed, then ϕ is automatically (F,F′)-harmonic;

• (2) if M is complete with Vol(M)=∞ and E˜B,2(ϕ)<∞, then ϕ is (F,F′)-harmonic.

• (3) If M is complete with EB(ϕ)<∞ and E˜B,2(ϕ)<∞, then ϕ is (F,F′) -harmonic.

Remark 1.3. On foliations, there is another kinds of harmonic map, called transversally harmonic map, which is a solution of the Eular-Lagrange equation τb(ϕ)=0 [15]. Also, the transversally biharmonic map is defined [4], which is not a critical point of the bienergy E˜B,2(ϕ). Two definitions for harmonic maps are equivalent when the foliation is minimal. The generalized Chen's conjectures for transversally biharmonic map have been proved in [11, 13].

Let (M,g,ℱ) be a foliated Riemannian manifold of dimension n with a foliation ℱ of codimension q(=n−p) and a bundle-like metric g with respect to ℱ [18, 23]. Let Q=TM/Tℱ be the normal bundle of ℱ, where Tℱ is the tangent bundle of ℱ. Let gQ be the induced metric by g on Q, that is, gQ=σ*(g|Tℱ⊥), where σ:Q→Tℱ⊥ is the cnonical bundle isomorphism. Then gQ is the holonomy invariant metric on Q, meaning that LXgQ=0 for X∈Tℱ, where LX is the transverse Lie derivative with respect to X. Let ∇Q be the transverse Levi-Civita connection on the normal bundle Q [23, 24] and R^Q be the transversal curvature tensor of ∇Q≡∇, which is defined by RQ(X,Y)=[∇X,∇Y]−∇[X,Y] for any X,Y∈ΓTM. Let KQ and RicQ be the transversal sectional curvature and transversal Ricci operator with respect to ∇, respectively. Let ΩBr(ℱ) be the space of all basic r-forms, i.e., ω∈ΩBr(ℱ) if and only if i(X)ω=0 and LXω=0 for any X∈ΓTℱ, where i(X) is the interior product. Then Ω*(M)=ΩB*(ℱ)⊕ΩB*(ℱ)⊥ [1]. It is well known that κB is closed, i.e., dκB=0, where κB is the basic part of the mean curvature form κ [1, 20] . Let *¯:ΩBr(F)→ΩBq−r(F) be the star operator given by

*¯ω=(−1)(n−q)(q−r)*(ω∧χF), ω∈ΩBr(F),

where χF is the characteristic form of F and * is the Hodge star operator associated to g. Let 〈⋅,⋅〉 be the pointwise inner product on ΩBr(F), which is given by

〈ω1,ω2〉ν=ω1∧*¯ω2,

where ν is the transversal volume form such that *ν=χF. Let δB:ΩBr(F)→ΩBr−1(F) be the operator defined by

δBω=(−1)q(r+1)+1*¯(dB−κB∧)*¯ω,

where dB=d|ΩB*(F). It is well known [22] that δB is the formal adjoint of d_B with respect to the global inner product. That is,

∫M 〈dω1 ,ω2 〉μM=∫M 〈ω1 ,δB ω2 〉μM

for any compactly supported basic forms ω1 and ω2, where μM=ν∧χF is the volume element.

There exists a bundle-like metric such that the mean curvature form satisfies δBκB=0 on compact manifolds [5, 16, 17]. The basic Laplacian ΔB acting on ΩB*(F) is given by

ΔB=dBδB+δBdB.

Now we define the bundle map AY:ΓQ→ΓQ for any Y ∈ TM by

(2.1)AYs=LYs−∇Ys,

where LYs=π[Y,Ys] for π(Ys)=s. It is well-known [14] that for any infitesimal automorphism Y (that is, [Y,Z]∈ΓTF for all Z∈ΓTF [14])

AYs=−∇Ysπ(Y),

where π:TM→Q is the natural projection and Y_s is the vector field such that π(Ys)=s. So A_Y depends only on Y¯=π(Y) and is a linear operator. Moreover, A_Y extends in an obvious way to tensors of any type on Q [14]. Then we have the generalized Weitzenböck formula on ΩB*(F) [10]: for any ω∈ΩBr(F),

(2.2)ΔBω=∇tr*∇trω+F(ω)+AκB♯ω,

where F(ω)=∑ a,bθa∧i(Eb)RQ(Eb,Ea)ω and

(2.3)∇tr*∇trω=−∑a∇E a ,E a 2ω+∇κB♯ω.

The operator ∇tr*∇tr is positive definite and formally self adjoint on the space of basic forms [10]. If ω is a basic 1-form, then F(ω)♯=RicQ(ω♯). Now, we recall the transversal divergence theorem on a foliated Riemannian manifold for later use.

Theorem 2.1. [26] Let (M,g,F) be a closed, oriented Riemannian manifold with a transversally oriented foliation F and a bundle-like metric g with respect to F. Then for a transversal infinitesimal automorphism X,

∫M div∇ (π(X))μM=∫M gQ (π(X),κB♯)μM,

where div∇s denotes the transversal divergence of s with respect to the connection ∇.

Let ϕ:(M,g,ℱ)→(M′,g′,ℱ′) be a smooth foliated map, i.e., dϕ(Tℱ)⊂Tℱ′, and ΩBr(E)=ΩBr(ℱ)⊗E be the space of E-valued basic r-forms, where E=ϕ−1Q′ is the pull-back bundle on M. We define dTϕ:Q→Q′ by

dTϕ:=π′∘dϕ∘σ.

Trivially, dTϕ∈ΩB1(E). Let ∇ϕ and ∇˜ be the connections on E and Q*⊗E, respectively. Then a foliated map ϕ:(M,g,F)→(M′,g′,F′) is called transversally totally geodesic if it satisfies

(3.1)∇˜trdTϕ=0,

where (∇˜trdTϕ)(X,Y)=(∇˜XdTϕ)(Y) for any X,Y∈ΓQ. Note that if ϕ:(M,g,F)→(M′,g′,F′) is transversally totally geodesic with dϕ(Q)⊂Q′, then, for any transversal geodesic γ in M, ϕ∘γ is also transversal geodesic. From now on, we use ∇ instead of all induced connections if we have no confusion. We define d∇:ΩBr(E)→ΩBr+1(E) by

(3.2)d∇(ω⊗s)=dBω⊗s+(−1)rω∧∇s

for any s∈ΓE and ω∈ΩBr(F). Let δ∇ be a formal adjoint of d∇ with respect to the inner product. Note that

(3.3)d∇(dTϕ)=0, δ∇dTϕ=−τb(ϕ)+dTϕ(κB♯),

where τb(ϕ) is the transversal tension field of ϕ defined by

(3.4)τb(ϕ):=trQ(∇trdTϕ).

The Laplacian Δ on ΩB*(E) is defined by

Δ=d∇δ∇+δ∇d∇.

Moreover, the operator A_X is extended to ΩBr(E) as follows:

AXΨ=LXΨ−∇XΨ,

where LX=d∇i(X)+i(X)d∇ for any X∈ΓTM and i(X)(ω⊗s)=i(X)ω⊗s. Hence Ψ∈ΩB*(E) if and only if i(X)Ψ=0 and LXΨ=0 for all X∈ΓTF.

Then the generalized Weitzenböck type formula (2.2) is extended to ΩB*(E) as follows [12]: for any Ψ∈ΩBr(E),

(3.5)ΔΨ=∇tr*∇trΨ+AκB♯Ψ+F(Ψ),

where ∇tr*∇tr is the operator induced from (2.3) and F(Ψ)=∑ a,b=1qθa∧i(Eb)R(Eb,Ea)Ψ. Moreover, we have that for any Ψ∈ΩBr(E),

(3.6)12ΔB|Ψ|2=〈ΔΨ,Ψ〉−|∇trΨ|2−〈AκB♯Ψ,Ψ〉−〈F(Ψ),Ψ〉.

3.1. (F,F′)-harmonic maps

About this section, see [6]. Let Ω be a compact domain of M. Then the transversal energy functional of ϕ on Ω is defined by

(3.7)EB(ϕ;Ω)=12∫Ω|dTϕ|2μM.

Then Dragomir and Tommasoli [6] defined (F,F′)-harmonic if ϕ is a critical point of the transversal energy functional EB(ϕ). Also, we obtain the first variational formula [6, 12]

(3.8)ddtEB(ϕt;Ω)|t=0=−∫Ω 〈τ˜b(ϕ),V〉μM,

where V=dϕtdt|t=0 is the normal variation vector field of a foliated variation {ϕt} of ϕ and

(3.9)τ˜b(ϕ):=τb(ϕ)−dTϕ(κB♯).

From (3.8), we have the following [6].

Proposition 3.1. A foliated map ϕ is (F,F′)-harmonic map if and only if τ˜b(ϕ)=0.

Remark 3.2. (1) If ϕ:M→ℝ is a basic function, then τ˜b(ϕ)=−ΔBϕ. So (F,F′)-harmonic map is a generalization of a basic harmonic function.

(2) On foliated manifold, there is another kinds of harmonic map, transvesally harmonic map, which is a solution of the Euler-Lagrange equation τb(ϕ)=0 by Konderak and Wolak [15]. But the transversally harmonic map is not a critical point of the energy functional EB(ϕ). Two definitions are equivalent when the foliation is minimal.

Now, we define the transversal Jacobi operator JϕT:Γϕ−1Q′→Γϕ−1Q′ by

(3.10)JϕT(V)=∇tr*∇trV−trQR Q′(V,dTϕ)dTϕ.

Then JϕT is a formally self-adjoint operator. That is, for any V,W∈Γϕ−1Q′,

(3.11)∫M 〈JϕT(V),W〉μM=∫M 〈V,JϕT(W)〉μM.

Also, we have the second variation formula for the transversal energy functional EB(ϕ).

Theorem 3.3. ([6], The second variation formula) Let ϕ:(M,g,F)→(M′,g′,F′) be a (F,F′)-harmonic map and let {ϕs,t} be the foliated variation of ϕ supported in a compact domain Ω. Then

(3.12)∂2∂s∂tEB(ϕs,t;Ω)|(s,t)=(0,0)=∫Ω 〈J ϕT(V),W〉μM,

where V and W are the variation vector fields of ϕs,t.

Proof. Let V=∂ϕs,t∂s|(s,t)=(0,0) and W=∂ϕs,t∂t|(s,t)=(0,0) be the variation vector fields of ϕs,t. Let Φ:M×(−ϵ,ϵ)×(−ϵ,ϵ)→M′ be a smooth map, which is defined by Φ(x,s,t)=ϕs,t(x). Let ∇Φ be the pull-back connection on Φ−1Q′. It is trivial that [X,∂∂t]=[X,∂∂s]=0 for any vector field X∈TM. From (3.8), we have

∂2∂s∂tEB(ϕs,t;Ω)=−∫Ω 〈 ∂ 2 ϕ s,t ∂s∂t ,τ˜b(ϕs,t)〉μM−∫Ω 〈 ∂ ϕ s,t ∂t ,∇ ∂ ∂s Φ τ˜b(ϕs,t)〉μM.

At (s,t)=(0,0), the first term vanishes because of τ˜b(ϕ)=0. So

(3.13)∂2∂s∂tEB(ϕs,t;Ω)|(s,t)=(0,0)=−∫Ω 〈W,∇ ∂ ∂s Φ τ˜b(ϕs,t)|(s,t)=(0,0)〉μM.

At x∈ M, by a straight calculation, we have

(3.14)∇∂∂sΦτ˜b(ϕs,t)=∑a∇E a Φ∇E aΦdΦ(∂∂s)−∇κB♯ΦdΦ(∂∂s)+∑aRΦ(ddt,Ea)dΦ(Ea).

Hence at (s,t)=(0,0), we have

∇∂∂sΦτ˜b(ϕs,t)|(s,t)=(0,0)=−∇tr*∇trV+trQR Q′(V,dTϕ)dTϕ.

That is, we have

(3.15)∇∂∂sΦτ˜b(ϕs,t)|(s,t)=(0,0)=−JϕT(V).

Hence the proof of (3.12) follows from (3.13) and (3.15).

Now, we define the basic Hessian HessϕT of ϕ by

(3.16)HessϕT(V,W)=∫M 〈J ϕ T (V),W〉μM.

Then HessϕT(V,W)=HessϕT(W,V) for any V,W∈ϕ−1Q′. If HessϕT is positive semi-definite, that is, HessϕT(V,V)≥0 for any normal vecor field V along ϕ, then ϕ is said to be weakly stable. Hence we have the following corollary.

Corollary 3.4. ([6], Stability) Let M be a closed Riemannian manifold and M' be of non-positive transversal sectional curvature. Then any (F,F′)-harmonic map ϕ:(M,F)→(M′,F′) is weakly stable.

Remark 3.5. For the stability of transversally harmonic map (that is, τb(ϕ)=0), see [11, Corollary 4.6]. In fact, under the same assumption, a transversally harmonic map is transversally f-stable, that is, ∫M 〈(JϕT−∇κB♯ )V,V〉e−fμM≥0, where f is a basic function such that κB=−df.

3.2. (F,F′)-biharmonic maps

We define the transversal bienergy functional E˜B,2(ϕ) on a compact domain Ω by

(3.17)E˜B,2(ϕ;Ω):=12∫Ω|τ˜b(ϕ)|2μM.

Definition 3.6. A foliated map ϕ:(M,g,F)→(M′,g′,F′) is said to be (F,F′)-biharmonic map if ϕ is a critical point of the transversal bienergy functional E˜B,2(ϕ).

Theorem 3.7. (The first variation formula) For a foliated map ϕ,

(3.18)ddtE˜B,2(ϕt;Ω)|t=0=−∫Ω 〈J ϕT ( τ ˜ b(ϕ)),V〉μM,

where V=dϕtdt|t=0 is the variation vector field of a foliated variation ϕ_t of ϕ.

Proof. Let Φ:M×(−ϵ,ϵ)→M′ be a smooth map, which is defined by Φ(x,t)=ϕt(x). Let ∇Φ be the pull-back connection on Φ−1Q′. It is trivial that [X,∂∂t]=0 for any vector field X ∈ TM. From (3.17), we have

(3.19)ddtE˜B,2(ϕt;Ω)|t=0=∫Ω 〈∇ d dt Φ τ ˜ b(ϕt)|t=0,τ˜b(ϕ)〉μM.

From (3.11), (3.15) and (3.19), we finish the proof.

From the first variation formula for the transversal bienergy functional, we know the following fact.

Proposition 3.8. A (F,F′)-biharmonic map ϕ is a solution of the following equation

(3.20)(τ˜2)b(ϕ):=JϕT(τ˜b(ϕ))=0.

Here (τ˜2)b(ϕ) is called the (F,F′)-bitension field of ϕ.

Remark 3.9. (1) From Remark 3.2, if ϕ is a basic function on M, then

(τ˜2)b(ϕ)=JϕT(τ˜b(ϕ))=−JϕT(ΔBϕ)=−∇tr*∇tr(ΔBϕ)=ΔB2ϕ.

So (F,F′)-biharmonic map is a generalization of basic biharmonic function.

(2) A (F,F′)-harmonic map is trivially (F,F′)-biharmonic map.

(3) There is another kinds of biharmonic map on foliations, called transversally biharmonic map, which is a solution of (τ2)b(ϕ):=JϕT(τb(ϕ))−∇κB♯τb(ϕ)=0 [11].

Actually, transversally biharmonic map is a critical point of the transversal f-bienergy functional E2,f(ϕ), which is defined by

E2,f(ϕ)=12∫M|τb(ϕ)|2e−fμM,

where f is a solution of κB=−df.

3.3. Generalized Chen's conjecture

Now, we consider the generalized Chen's conjecture for (F,F′)-biharmonic maps.

Theorem 3.10. Let (M,g,F) be a foliated Riemannian manifold and let (M′,g′,F′) be of non-positive transversal sectional curvature, that is, K Q′≤0. Let ϕ:M→M′ be a (F,F′)-biharmonic map. Then

• (1) if M is closed, then ϕ is automatically (F,F′)-harmonic;

• (2) if M is complete with Vol(M)=∞ and E˜B,2(ϕ)<∞, then ϕ is (F,F′)-harmonic;

• (3) If M is complete with EB(ϕ)<∞ and E˜B,2(ϕ)<∞, then ϕ is (F,F′) -harmonic.

Let ϕ:M→M′ be a (F,F′)-biharmonic map. Then from (3.20)

(3.21)(∇trϕ)*(∇trϕ)τ˜b(ϕ)−∑aR Q′ (τ˜b(ϕ),dTϕ(Ea))dTϕ(Ea)=0,

where {Ea} be a local orthonomal basic frame of Q. From the generalized Weitzenbock formula (3.5) and (3.6), we have

(3.22)12ΔB|τ˜b(ϕ)|2=〈∇tr*∇trτ˜b(ϕ),τ˜b(ϕ)〉−|∇trτ˜b(ϕ)|2.

Hence from (3.21), we get

12ΔB|τ˜b(ϕ)|2=−|∇trτ˜b(ϕ)|2+∑a〈RQ′(τ˜b(ϕ),dTϕ(Ea))dTϕ(Ea),τ˜b(ϕ)〉.

That is,

(3.23)|τ˜b(ϕ)|ΔB|τ˜b(ϕ)| = |dB|τ˜b(ϕ)||2−|∇trτ˜b(ϕ)|2        +∑a〈RQ′ (τ˜b(ϕ),dTϕ(Ea))dTϕ(Ea),τ˜b(ϕ)〉.

By the Kato's inequality, that is, |∇trτ˜b(ϕ)|≥|dB|τ˜b(ϕ)||, and K Q′≤0, we have

(3.24)12ΔB|τ˜b(ϕ)|≤0.

That is, |τ˜b(ϕ)| is basic subharmonic.

(1) If M is closed, then |τ˜b(ϕ)| is trivially constant. From (3.23), we have that for all a,

(3.25)∇Eaτ˜b(ϕ)=0.

Now, we define the normal vector field Y by

Y=∑a〈dTϕ(E a),τ˜b(ϕ)〉Ea.

Then from (3.25), we have

(3.26)div∇(Y)=∑a〈∇E a Y,E a〉=〈τb(ϕ),τ˜b(ϕ)〉.

So by integrating (3.26) and by using the transversal divergence theorem (Theorem 2.1), we get

(3.27)∫M|τ˜b(ϕ)|2μM=0,

which implies that τ˜b(ϕ)=0, that is, ϕ is the (F,F′)-harmonic map.

(2) Let M be a complete Riemannian manifold. Note that for any basic 1-form ω, it is trivial that δBω=δω and so ΔBf=Δf for any basic function f. Hence by the Yau's maximum principle [25, Theorem 3], we have following lemma.

Lemma 3.11. If a nonnegative basic function f is basic-subharmonic, that is, ΔBf≤0, with ∫M fp<∞ (p>1), then f is constant.

Since E˜B,2(ϕ)<∞, by (3.24) and Lemma 3.11, |τ˜b(ϕ)| is constant. Moreover, since Vol(M)=∞, ∫M|τ˜B(ϕ)|2μM<∞ implies τ˜b(ϕ)=0, that is, ϕ is (F,F′)-harmonic.

(3) Now we define a basic 1-form ω on M by

(3.28)ω(X)=〈dTϕ(X),τ˜b(ϕ)〉

for any normal vector field X. By using the Schwartz inequality, we get

∫M|ω|μM=∫M(∑a|ω(Ea)|2)12μM    =∫M(∑a|〈dTϕ(Ea),τ˜b(ϕ)〉|2)12μM    ≤∫M|dTϕ|τ˜b(ϕ)μM    ≤(∫M|dTϕ|2μM)12(∫M|τ˜b(ϕ)|2μM)12    =2EB(ϕ)EB,2(ϕ)<∞.

On the other hand, by a straight calculation, we know that

(3.29)δBω=−|τ˜b(ϕ)|2.

Since ∫M|ω|μM<∞ and ∫M (δB )ωμM=−E˜B,2(∞)<∞, by the Gaffney's theorem [8], we know that

(3.30)∫M|τ˜b(ϕ)|2μM=−∫M (δB ω)μM=−∫M (δω)μM=0.

Hence τ˜b(ϕ)=0, that is, ϕ is (F,F′)-harmonic.

Remark 3.12. Note that for transversally biharmonic map, we need some conditions that the transversal Ricci curvature of M is nonnegative and positive at some point (cf. [11, Theorem 6.5]).

Now, we study the second variation formula for the transversal bienergy functional E˜B,2(ϕ).

Theorem 3.13. (The second variation formula) For a foliated map ϕ:(M,g,F)→(M′,g′,F′), we have

d2dt2E˜B,2(ϕt;Ω)|t=0=−∫Ω 〈∇V V, ( τ ˜ 2)b (ϕ)〉μM+∫Ω |JϕT(V)|2μM        −∫Ω 〈RQ′ (V,τ˜b(ϕ))τ˜b(ϕ),V〉μM        −4∫M 〈RQ′ (∇trV,τ˜b(ϕ))dTϕ,V〉μM        +∫Ω 〈(∇τ˜b(ϕ)R Q′ )(V,dTϕ)dTϕ,V〉μM        +2∫Ω 〈(∇trR Q′ )(dTϕ,V)τ˜b(ϕ),V〉μM,

where V=dϕtdt|t=0 is the normal variation vector field of {ϕt}.

Proof Let Φ:M×(−ϵ,ϵ)→M′ be a smooth map, which is defined by Φ(x,t)=ϕt(x). Let ∇^𝚽 be the pull-back connection on Φ−1Q′. It is trivial that [X,∂∂t]=0 for any vector field X∈ TM. From definition, we have

d2dt2E˜B,2(ϕt;Ω)=∫Ω 〈∇ d dt Φ∇ d dt Φ τ ˜ b(ϕt),τ˜b(ϕt)〉μM+∫Ω|∇ddtΦτ˜b(ϕt)|2μM.

Let {Ea} be a local orthonormal basic frame on Q such that ∇ΦEa=0 at x∈ M. From (3.14), we have

∇ddtΦ∇ddtΦτ˜b(ϕt)=∑a∇E aΦ∇E aΦ∇ddtΦdΦ(ddt)−∇κB♯Φ∇ddtΦdΦ(ddt)+RΦ(κB♯,ddt)dΦ(ddt)      +∑a∇E aΦRΦ(ddt,Ea)dΦ(ddt)+∑a∇ddtΦRΦ(ddt,Ea)dΦ(Ea)      +∑aRΦ(ddt,Ea)∇E aΦdΦ(ddt).

At t=0, since dΦ(ddt)|t=0=dϕtdt|t=0=V, we have

∇ddtΦ∇ddtΦτ˜b(ϕt)|t=0=∑a∇E a∇E a∇VV−∇κB♯∇VV+RQ′(dTϕ(κB♯),V)V      +∑a∇E aRQ′(V,dTϕ(Ea))V      +∑a∇VRQ′(V,dTϕ(Ea))dTϕ(Ea)      +∑a RQ′ (V,dTϕ(Ea))∇E aV.

By a straight calculation together with the Bianchi identities, we have

∑a∇E aRQ′(V,dTϕ(Ea))V=∑a(∇E a RQ′ )(V,dTϕ(E a))V+RQ′ (V,τb(ϕ))V          +2∑a RQ′ (V,dTϕ(Ea))∇E aV          −∑a RQ′ (V,∇E aV)dTϕ(Ea)

and

∑a∇VRQ′(V,dTϕ(Ea))dTϕ(Ea)=∑a(∇V RQ′ )(V,dTϕ(E a))dTϕ(Ea)            +∑a RQ′ (∇VV,dTϕ(Ea))dTϕ(Ea)            +∑a RQ′ (V,∇E aV)dTϕ(Ea)            +∑a RQ′ (V,dTϕ(Ea))∇E aV.

By summing the above equations, we have

∇ddt∇ddtτ˜b(ϕt)|t=0=−JϕT(∇VV)+RQ′(V,τ˜b(ϕ))V      +∑a(∇V RQ′ )(V,dTϕ(E a))dTϕ(Ea)      +∑a(∇E a RQ′ )(V,dTϕ(E a))V      +4∑a RQ′ (V,dTϕ(Ea))∇E aV.

Then by integrating, we get

∫Ω 〈∇ d dtΦ∇ d dtΦ τ˜b (ϕt)|t=0,τ˜b(ϕ)〉=−∫Ω 〈J ϕT(∇VV),τ˜b(ϕ)〉          +∫Ω 〈RQ′(V,τ˜b(ϕ))V,τ˜b(ϕ)〉          +∑a∫Ω 〈(∇VR Q′)(V,dTϕ(Ea))dTϕ(Ea),τ˜b(ϕ)〉          +∑a∫Ω 〈(∇E a R Q′)(V,dTϕ(Ea))V,τ˜b(ϕ)〉          +4∑a∫Ω 〈RQ′ (V,dTϕ(Ea))∇E a V,τ˜b(ϕ)〉.

From the second Bianchi identity, we get

〈(∇VR Q′)(V,dTϕ(Ea))dTϕ(Ea),τ˜b(ϕ)〉=〈(∇EaR Q′)(V,dTϕ(Ea))V,τ˜b(ϕ)〉              +〈(∇τ˜b(ϕ)R Q′)(V,dTϕ(Ea))dTϕ(Ea),V〉.

From the above equation, we get

∫Ω 〈∇ d dtΦ∇ d dtΦ τ˜b (ϕt)|t=0,τ˜b(ϕ)〉= −∫Ω 〈J ϕT(∇VV),τ˜b(ϕ)〉+∫Ω 〈RQ′(V,τ˜b(ϕ))V,τ˜b(ϕ)〉          +∑a∫Ω 〈(∇τ˜b(ϕ)R Q′)(V,dTϕ(Ea))dTϕ(Ea),V〉          +2∑a∫Ω 〈(∇E a R Q′)(V,dTϕ(Ea))V,τ˜b(ϕ)〉          +4∑a∫Ω 〈RQ′ (V,dTϕ(Ea))∇E a V,τ˜b(ϕ)〉.

From the above equation and (3.15), by using the curvature properties and self-adjointness of JϕT, the proof follows.

Definition 3.14. A (F,F′)-biharmonic map ϕ:(M,g,F)→(M′,g′,F′) is said to be weakly stable if d2dt2E˜B,2(ϕt)|t=0≥0.

Now, we consider the generalized Chen's conjecture for (F,F′)-biharmonic map when the transversal sectional curvature of M' is positive, that is, K Q′>0. In case of K Q′≤0, see Theorem 3.10.

Let us recall the transversal stress-energy tensor ST(ϕ) of ϕ [4, 11]:

(3.31)ST(ϕ)=12|dTϕ|2gQ−ϕ*g Q′ .

Note that for any vector field X∈ΓQ,

(3.32)(div∇ST(ϕ))(X)=−〈τb(ϕ),dTϕ(X)〉.

If div∇ST(ϕ)=0, then we say that ϕ satisfies the transverse conservation law [4]. If there exists a basic function λ2 such that ϕ*g Q′ =λ2gQ, then ϕ is called a transversally weakly conformal map. In the case of λ being nonzero constant, ϕ is called a transversally homothetic map. Hence we have the following propositions.

Proposition 3.15. [7]} Let ϕ:(M,g,F)→(M′,g′,F′) be a transversally weakly conformal map with codim(F)>2. Then ϕ is transversally homothetic if and only if ϕ satisfies the transverse conservation law.

Theorem 3.16. Let (M,g,F) be a closed foliated Riemannian manifold and (M′,g′,F′) be a foliated Riemannian manifold with a positive constant transversal sectional curvature K Q′. Let ϕ:M→M′ be a (F,F′)-biharmonic map such that ϕ is transversally weakly stable and satisfies the transverse conservation law. If F is minimal or ϕ is transversally weakly conformal with codimF>2, then ϕ is (F,F′)-harmonic.

Proof. Let ϕ:M→M′ be a (F,F′)-biharmonic map, that is, (τ˜2)b(ϕ)=0.

Let K Q′=c>0, where c is a positive constant. Then for any X,Y,Z∈ΓQ′

(3.33)R Q′(X,Y)Z=c{〈Y,Z〉X−〈X,Z〉Y}.

So (∇XR Q′)(Y,Z)=0. Hence if we take V=τ˜b(ϕ) in Theorem 3.13, then from (3.33)

d2dt2E˜B,2(ϕt)|t=0=−4∫M 〈RQ′ (∇trτ˜b(ϕ),τ˜b)dTϕ,τ˜b(ϕ)〉μM      =−4c∫M 〈 τ ˜ b (ϕ),dTϕ〉〈∇trτ˜b(ϕ),τ˜b(ϕ)〉μM      +4c∫M 〈 d T ϕ, ∇ tr τ ˜ b (ϕ)〉|τ˜b(ϕ)|2μM      =−4c∫M 〈τ b (ϕ),τ˜b(ϕ)〉|τ˜b(ϕ)|2μM      +4c∑a∫M Ea (〈dTϕ(Ea),τ˜b(ϕ)〉|τ˜b(ϕ)|2)μM      −12c∑a∫ 〈 d T ϕ(Ea ), τ ˜ b( ϕ)〉〈∇ E aτ˜b(ϕ),τ˜b(ϕ)〉μM.

If we choose a normal vector field X as

〈X,Y〉=〈τ˜b(ϕ),dTϕ(Y)〉|τ˜b(ϕ)|2

for any normal vector field Y, then

div∇X=∑aE a(〈τ˜b(ϕ),dTϕ(Ea)〉|τ˜b(ϕ)|2).

Hence by the transversal divergence theorem, we have

∫ ∑aE a(〈dTϕ(Ea),τ˜b(ϕ)〉|τ˜b(ϕ)|2)μM=∫div∇ (X)μM=∫〈X,κB♯〉μM              =∫〈 d T ϕ(κB♯),τ˜b(ϕ)〉|τ˜b(ϕ)|2μM.

Combining the above equations, we have

(3.34)d2dt2E˜B,2(ϕt)|t=0=−4c∫M |τ˜b(ϕ)|4μM      −12c∑a∫M 〈 τ ˜ b( ϕ),dTϕ(Ea)〉〈∇ E aτ˜b(ϕ),τ˜b(ϕ)〉μM.

Since ϕ satisfies the transverse conservation law, that is, (div∇ST(ϕ))(X)=0 for any X, we have

〈τb(ϕ),dTϕ(Ea)〉=(div∇ST(ϕ))(Ea)=0.

Moreover, since ϕ is transversally weakly conforml, from Proposition 3.15, ϕ is transversally homothetic. Hence

∑a〈dTϕ(κB♯),dTϕ(Ea)〉〈∇E aτ˜b(ϕ),τ˜b(ϕ)〉=α〈∇κB♯τ˜b(ϕ),τ˜b(ϕ)〉

for some constant α. So if we choose the bundle-like metric such that δBκB=0, then

∫M ∑a 〈τ˜b (ϕ),dTϕ(Ea)〉〈∇Ea τ˜b(ϕ),τ˜b(ϕ)〉μM=−α∫M 〈 ∇ κB♯ τ ˜ b (ϕ),τ˜b(ϕ)〉μM=−α2∫M 〈δB κB ,| τ ˜ b (ϕ)|〉μM=0.

Hence from (3.34), we have

(3.35)d2dt2E˜B,2(ϕt)|t=0=−4c∫ |τ˜b(ϕ)|4μM.

In case F is minimal, (3.35) also holds. Hence since ϕ is weakly stable and c>0, we have τ˜b(ϕ)=0, that is, ϕ is (F,F′)-harmonic.

Remark 3.17. The generalized Chen's conjectures for the transversally biharmonic map have been studied in [11, 13] under some additional conditions such that the transversal Ricci curvature of M is nonnegative.