Zhejiang University · 2013-05-17 · f-BIHARMONIC MAPS BETWEEN DOUBLY WARPED PRODUCT MANIFOLDS...

f -BIHARMONIC MAPS BETWEEN DOUBLY WARPED PRODUCTMANIFOLDS

WEI-JUN LU

A. In this paper, by applying the first variation formula of f -bi-energy given in[OND], we study f -biharmonic maps between doubly warped product manifolds M ×(µ,λ)N. Under imposing existence condition concerning proper f -biharmonic maps, we derivef -biharmonicity’s characteristic equations for the inclusion maps: iy0 : (M, g)→ (M ×(µ,λ)

N, g), ix0 : (N, h) → (M ×(µ,λ) N, g) and the product maps: Ψ = IdM × ϕN : M ×(µ,λ) N →M × N, Ψ = ˜ϕM × IdN : M ×(µ,λ) N → M × N with ϕMϕN being a harmonic map.

1. I

f -Biharmonic maps between Riemannian manifolds, as a generalization of biharmonicmaps and f -harmonic maps, were first introduced by Ouakkas etc. in 2010 [OND]. Onthe one hand, f -harmonic maps between Riemannian manifolds ( as a generalization ofharmonic maps ) were first introduced and studied by Lichnerowicz in [L](see also Section10.20 in Eells-Lemaires report [EL]). The study of f-harmonic maps comes from a physicalmotivation, since in physics, f -harmonic map can be viewed as stationary solution of theinhomogeneous Heisenberg spin system ( see [LW]). On the other hand, the researchin the field of Riemannian geometry in the last 10-15 years was partly characterized bythe study of some fourth order partial differential equations. Whether the origin of theseequations is analytic, as in the case of the Paneitz operator, or geometric, as in the caseof Willmore surfaces, they represent a generalization of the concept of harmonic map. Anatural extension of the harmonic maps was suggested by Eells and Sampson themselvesin [ES]. The first results in this field were obtained by Jiang in 1986 (see [J]. He derived theEuler-Lagrange equation of the bienergy functional given by the vanishing of the bitensionfield

τ2(φ) = trace(∇φ. ∇

φ. τ(φ) − ∇

φM∇. .τ(φ))− traceg(RN(dφ, τ(φ))dφ)) = 0,

where φ : (M, g) → (N, h) is a smooth map between two Riemannian manifolds. Theessential difference between both f -harmonic maps and biharmonic maps is that the energyfunctional

∫ΩbM e(φ)dvg, where in former case e(φ) = 1

2 f |dφ|2 with positive function f ∈C∞(M), while in the latter e(φ) = 1

2 |τ(φ)|2 with the tension field τ(φ) = traceg∇dφ. f -

Biharmonic maps consider to integrate the above two generalized harmonic maps and usethe energy density e(φ) = 1

2 |τ f (φ)|2 with the f -tension field τ f (φ) = f τ(φ) + dφ(grad f ).Since any harmonic map is f -biharmonic, we are interested in non-harmonic f -biharmonicmaps which are called proper biharmonic.

The concept of warped product plays an important role in differential geometry as wellas in theoretical physics. The notion of warped metric was first introduced by Bishopand O’Neill in [BO]. Since then, in Riemannian geometry, warped products have offered

Date: July 12, 2011.2000 Mathematics Subject Classification. 58E20; 53C12.Key words and phrases. f -Harmonic; f -biharmonic; doubly warped product manifold; product map.

1

2 WEI-JUN LU

new examples of Riemannian manifolds with special curvature properties. Beem, Ehrlichand Powell pointed out that many exact solutions to Einstein’s field equation can be ex-pressed in terms of Lorentzian warped products in [BEP]. Furthermore, Beem and Ehrlichconcluded that causality and completeness of warped products can be related to causalityand completeness of components of warped products in [BE]. O’Neill discussed warpedproducts and explored curvature formulas of warped products in terms of curvatures ofcomponents of warped products in [ON]. He also examined Robertson-Walker, static,Schwarschild and Kruskal space-times as warped products.

In general, doubly warped products (M × N, µ2g + λ2h) can be considered as a general-ization of singly warped products (M×N, g+λ2h) ( see Definitions 3.1). Beem and Powellconsidered these products for Lorentzian manifolds in [BP]. Allison considered causal-ity and global hyperbolicity of doubly warped products in [A1] and null pseudocovexityof Lorentzian doubly warped products in [A2]. Conformal properties of doubly warpedproducts are studied by Gebarowski (cf. [G] or references therein).

As an application of biharmonic maps, the biharmonic maps were studied from betweensingle warped product manifolds ([SE],[BMO]). The authors of [SE] investigated bihar-monicity of the inclusion iy0 : M → M ×(µ,λ) N rsp. ix0 : N → M ×(µ,λ) N of a Riemannianmanifold M rsp. N into the doubly warped product manifold M ×(µ,λ) N and of the projec-tion from M ×(µ,λ) N into the first factor M resp. the second factor N. Also, they gave twonew classes of proper biharmonic maps by using product of harmonic maps and doublywarping the metric in the domain or codomain Ψ : (M × N, g ⊕ h)→ (M × N, g ⊕ h).

Motivated by the mentioned works, we have extended these ideas to study f -harmonicby considering the situation of doubly warped products manifolds [Lu]. In this paper, weshall continue to study f -biharmonic in the same setting based on our previous work [Lu]and the work [SE]. Since the the f -bitension field τ2

f (φ) (see (2.3)) is rather complex whenφ is any one of the cases such as the inclusion map, the projection, the special productmap, the general product map so on, we only obtain some distinct characterization results.Especially, it is a bit pity that we cannot yet find a concrete example in terms of our presentunderstanding. However, we conjecture that the study of f -biharmonic map/morphism willbecome a inevitable trend in the future because it inherits each merits from biharmonic andf -harmonic map. This is a strong encouragement for the study of f -biharmonic maps.

The structure of this paper is as follows. In the second section we give some basicdefinitions on biharmonic maps, f -harmonic maps and f -biharmonic maps. In Section3, we give the definition of doubly warped product manifolds and the more explicit ex-pressions of the connection ∇ on doubly warped product , as well as give the expressionof curvature tensor R on doubly warped product M ×(µ,λ) N by referring to [PR]. Section4 is devoted to analyzing the conditions for both of the leaves x0 × N and M × y0 tobe f -biharmonic as a submanifold and we can’t show that both of the leaves x0 × Nand M × y0 can not be proper f -biharmonic as a submanifold of the doubly warpedproduct manifold M ×(µ,λ) N. In Section 5, we characterize a doubly warped product ac-cording to its projection map π1 : M ×(µ,λ) N → M and π2 : M ×(µ,λ) N → N being af -biharmonic maps for some particular functions. We consider the product of two har-monic maps Ψ : (M × N, g ⊕ h) → (M × N, g ⊕ h). By doubly warping the metric on thedomain or codomain we lose the harmonicity; nevertheless, under certain conditions on thetwo warping functions, the product mapsΨ = IdM × φN , Ψ = ˜φM × IdN and Ψ = IdM × φN

remain f -harmonic. In last section, we character the case under general product maps.

f -BIHARMONIC MAPS BETWEEN DOUBLY WARPED PRODUCT MANIFOLDS 3

2. f - f -

Recall that the energy of a smooth map φ : (M, g) → (N, h) between two Riemannianmanifolds is defined by integral E(φ) =

∫Ω

e(φ)dvg, for every compact domain Ω ⊂ Mwhere e(φ) = 1

2 |dφ|2 is energy density and φ is called harmonic if it’s a critical point of en-

ergy. From the first variation formula for the energy, the Euler-Lagrange equation is givenby the vanishing of the tension field τ(φ) = traceg∇dφ (see [ES]). As the generalizationsof harmonic maps, we now introduce the concepts of biharmonic maps and f -harmonicmaps

Definition 2.1. (i) Biharmonic maps φ : (M, g) → (N, h) between Riemannian manifoldsare critical points of the bienergy functional

E2(φ) =12

∫Ω

|τ(φ)|2dvg,

for any compact domain Ω ⊂ M.(ii) An f -harmonic map with a positive function f ∈ C∞(M) is a critical point of

f−energy

(2.1) E f (φ) =12

∫ΩbM

f |d(φ)|2dvg.

The Euler-Lagrange equations give the bitension field τ2(φ)( [J]) and the f -tension fieldequation τ f (φ) (see [C1], [OND],[Ou]), respectively,

(2.2)τ2(φ) = trace

(∇φ. ∇

φ. τ(φ) − ∇

φM∇. .τ(φ))− traceg(RN(dφ, τ(φ))dφ)) = 0,

τ f (φ) = f τ(φ) + dφ(grad f ) = 0.

A more natural generalization of f -harmonic maps and bi-harmonic maps is given byintegrating the square of the norm of the f -tension field, introduced recently in [OND].The authors of [OND] give the first and second variations of f -bi-energy functional. Butthey cannot give any example about f -biharmonic maps.

Definition 2.2. f -bi-energy functional of smooth map φ : (M, g)→ (N, h) is defined by

(2.3) E2f (φ) =

12

∫Ω

|τ f (φ)|2dV

for every compact domain Ω ⊂ M. A map φ is called f -biharmonic map if it the criticalpoint of f -bi-energy functional.

From the first variation, we obtain the Euler-Lagrange equation gives the f -bi-harmonicmap equation ( [OND])

(2.4) τ2f (φ) = −∆

2f τ f (φ) − f tracegRN(τ f (φ), dφ)dφ = 0

where∆2

f τ f (φ) = −traceg(∇φ f∇φτ f (φ) − f∇φ

∇..τ f (φ)

).

For an orthonormal frame eimi=1, we have

(2.5)traceg

(∇φ f∇φτ f (φ) − f∇φ

∇..τ f (φ)

)=

m∑i=1

(∇φei f∇φeiτ f (φ) − f∇φ

∇ei eiτ f (φ)

)=

m∑i=1

(f∇φei∇

φeiτ f (φ) − f∇φ

∇ei eiτ f (φ) − ∇

φgrad f τ f (φ)

)τ2

f (φ) is called the f -bi-tension field of φ.

4 WEI-JUN LU

Remark 2.3. From (2.5), the f -bi-tension field of φ can be simplified the following expres-sion

(2.6) τ2f (φ) = f [Trg(∇φ)2τ f (φ) + TrgRN(dφ, τ f (φ))dφ] − ∇

φgrad f τ f (φ),

where (∇φ)2 = ∇φ∇φ − ∇φ∇...

3. R

First we refer to [U] and give the definition of doubly warped product manifold.

Definition 3.1. Let (M, g) and (N, h) be Riemannian manifolds of dimensions m and n,respectively and let λ : M → (0,+∞) and µ : N → (0,+∞) be smooth functions. A doublywarped product (G, g) with warping functions λ and µ is a product manifold which is ofthe form G = M ×(µ,λ) N with the metric

g = µ2g ⊕ λ2h.

When µ ≡ 1 we have a (single) warped product M ×λ N with the warping function λ(x)

For the connection of doubly warped product manifold G = (M ×(µ,λ) N, g), by referringto [SE] or [PR], we have

Proposition 3.2. Let (M, g) and (N, h) be Riemannian manifolds with Levi-Civita connec-tions M∇ and N∇ , respectively and let ∇ and ∇ denote the Levi-Civita connections of theproduct manifold M×N and doubly warped product manifold G := M×(µ,λ) N, respectively,where λ : M → (0,∞) and µ : N → (0,∞) are smooth maps, g = µ2g ⊕ λ2h. Then we getthe Levi-Civita connection of doubly warped product manifold G as follows:

(3.1)

∇(X1,Y1)(X2,Y2) = ∇(X1,Y1)(X2,Y2) + 12λ2 X1(λ2)(01,Y2) + 1

2λ2 X2(λ2)(01,Y1)+ 1

2µ2 Y1(µ2)(X2, 02) + 12µ2 Y2(µ2)(X1, 02)

− 12 g(X1, X2)(01, grad µ2) − 1

2 h(Y1,Y2)(grad λ2, 02)= ( M∇X1 X2 +

12µ2 Y1(µ2)X2 +

12µ2 Y2(µ2)X1 −

12 h(Y1,Y2)grad λ2, 02)

+(01,N∇Y1 Y2 +

12λ2 X1(λ2)Y2 +

12λ2 X2(λ2)Y1 −

12 g(X1, X2)gradN µ

2)

for any (X1,Y1), (X2,Y2) ∈ Γ(TG), X1, X2 ∈ Γ(T M) and Y1,Y2 ∈ Γ(T N).

Proof. See the proof of Proposition 2.1 in [Lu].

Now we look at the curvature tensor of the warped product, we shall make optimal andgive a rewritten expression different from that in [PR, SE] by organizing all the terms afterthe order. Since apparently no explicit version is given in the literature as far I know, wegive our independent proof in the following.

Proposition 3.3. Let (M, g) and (N, h) be Riemannian manifolds with Levi-Civita connec-tions M∇ and N∇ , respectively and let ∇ and ∇ denote the Levi-civita connections of theproduct manifold M × N and doubly warped product manifold G := M ×(µ,λ) N, respec-tively. If R and R denote the curvature tensors of M × N and G, respectively, then for all


X = (X1,Y1), Y = (X2,Y2), Z = (X3,Y3) ∈ Γ(T M × N) ,we have the following relation

(3.2)

R(X,Y)Z = R(X,Y)Z+∑σ∈S(1,2) sign(σ)

[ λ

2

2µ2 h( N∇Yσ(1) gradλ2hµ

2 − 12µ2 Yσ(1)(µ2) gradλ2hµ

2, Y3)

− 14λ2 g(Yσ(1)(µ2) gradµ2gλ

2, X3) − |gradλ2hµ2 |2

4µ2 g(Xσ(1), X3)](Xσ(2), 02)

+[ µ2

2λ2 g( M∇Xσ(1) gradµ2gλ

2 − 12λ2 Xσ(1)(λ2) gradµ2gλ

2, X3)

− 14µ2 h(Xσ(1)(λ2) gradλ2hµ

2, Y3) −|gradµ2gλ

2 |2

4λ2 h(Yσ(1),Y3)](01, Yσ(2))+ 1

2 h(Yσ(1),Y3)( M∇Xσ(2) gradµ2gλ

2 − 12λ2 Xσ(2)(λ2) gradµ2gλ

2, 02)

+ 12 g(Xσ(1), X3)

(01,

N∇Yσ(2) gradλ2hµ2 − 1

2µ2 Yσ(2)(µ2) gradλ2hµ2)

+ 14λ2 g(Xσ(2), X3)

(Yσ(1)(µ2) gradµ2gλ

2, 02)

+ 14µ2 h(Yσ(2),Y3)

(01, Xσ(1)(λ2) gradλ2hµ

2),or

(3.3)

R(X,Y)Z = R(X,Y)Z+ 1

2µ2

∑σ∈S(1,2)

sign(σ)(Xσ(1), 02) ∧g [(01,N∇Yσ(2) gradλ2hµ

2

− 12µ2 Yσ(2)(µ2) gradλ2hµ

2 ) − ( 12λ2 Yσ(2)(µ2) gradµ2gλ

2, 02 ) ] ∧g ( X3,Y3 )+ 1

4µ4 |gradλ2hµ2|2[ (X2, 02) ∧g (X1, 02) ] ∧g ( X3,Y3 )

+ 12λ2

∑σ∈S(1,2)

sign(σ) (01, Yσ(1)) ∧g [( M∇Xσ(2) gradµ2gλ2

− 12λ2 Xσ(2)(λ2) gradµ2gλ

2, 02 ) − ( 01,1

2µ2 Xσ(2)(λ2) gradλ2hµ2 ) ]

∧g( X3,Y3 ) + 14λ4 |gradµ2gλ

2|2[ (01,Y2) ∧g (01,Y1) ] ∧g ( X3,Y3 ),

where the doubly wedge product (X ∧g Y) ∧g Z denotes g(X,Z)Y − g(Y,Z)X , and S(1, 2)denotes a permutation group generated by elements 1, 2,

sign(σ) =

1, (σ(1), σ(2)) = (1, 2);−1, (σ(1), σ(2)) = (2, 1).

4. f -

In this section, we consider the inclusion map of M

iy0 : (M, g)→ (M ×(µ,λ) N, g)x 7→ (x, y0)

at the point y0 ∈ N level in M ×(µ,λ) N and the inclusion map of N

ix0 : (N, h)→ (M ×(µ,λ) N, g)y 7→ (x0, y)

at the point x0 level in M×(µ,λ) N. By generalizing the methods in [SE] and [Lu], we obtainsome non-existence results for the f -harmonicity of inclusion maps iy0 of M and ix0 of Nunder the doubly warped product case.

Theorem 4.1. Let f : M → (0,+∞) be a smooth function. The inclusion map of themanifold (M, g) into the nontrivial (proper) doubly warped product manifold G = M×(µ,λ)Nis a proper f -harmonic map if and only if λ, µ, f satisfy

(4.1)f (Trg( M∇)2gradg f − m

8λ2 f |gradλ2hµ2|2)gradµ2gλ

2

+m2 f8λ2 |gradλ2hµ

2|2gradµ2gλ2 + RieM(gradg f )

− m2

4µ2 |gradλ2hµ2|2gradg f + 1

4µ2 |gradλ2hµ2|2gradg f − M∇gradg f gradg f = 0

6 WEI-JUN LU

and

(4.2)− 1

2 |gradg f |2gradλ2hµ2 + f (− 1

2 (∆g( f ) − M∇e j e j( f )) gradλ2hµ2

−m2 f

4 [ N∇gradλ2hµ2 gradλ2hµ

2 − 12µ2 f |gradλ2hµ

2|2gradλ2hµ2] ) = 0.

Proof. Let e jmj=1 be an orthonormal frame on (M, g). By using the tension field of iy0

τ(iy0 ) = Trg∇diy0 =m∑

j=1(∇

i−1y0

T Ge j diy0 )(e j)

=m∑

j=1(∇(e j,02)(e j, 02) − ( M∇e j e j, 02)

= −m2 (01, gradλ2hµ

2) (by (3.1))

and (2.2), the f-tension field of iy0 is

(4.3) τ f (iy0 ) = −m2

(01, f · gradλ2hµ2) + (gradg f , 02)

Since from (2.6) the f-biharmonic map of iy0 is

(4.4)

τ2f (iy0 ) = f [Trg(∇iy0 )2τ f (iy0 ) + TrgR(diy0 , τ f (iy0 ) )diy0 ] − ∇

iy0gradg f τ f (iy0 )

= fm∑

j=1[∇

iy0e j ∇

iy0e j − ∇

iy0M∇e j e j

]τ f (iy0 )

+R((e j, 02), τ f iy0

)(e j, 02) − ∇(gradg f ,0)diy0 (τ f (iy0 )),

we only need to calculus the following terms:

∇iy0e j τ f (iy0 ) = ∇(e j,0)(gradg f , 02 )= ( M∇e j gradg f , 0) − 1

2 g(e j, gradg f )(0, gradλ2hµ2)

= ( M∇e j gradg f , 02 ) − 12 e j( f )(01, gradλ2hµ

2) ),

(4.5)∇

iy0M∇e j e j

(τ f (iy0 )) = ∇( M∇e j e j,02)(gradg f , 02 )

= ( M∇ M∇e j e j gradg f , 02 ) − 12 ( M∇e j e j)( f )(01, gradλ2hµ

2) ),

(4.6) ∇(gradg f ,0)diy0 (τ f (iy0 )) = ( M∇gradg f gradg f , 02 ) − 12 |gradg f |2(01, gradλ2hµ

2) ),

(4.7)∇

iy0e j ∇

iy0e j τ f (iy0 ) = ∇(e j,0)( M∇e j gradg f , 02 )

= ( M∇e jM∇e j gradg f , 02 ) − 1

2 g(e j,M∇e j gradg f )(01, gradλ2hµ

2) )= ( M∇e j

M∇e j gradg f , 02 ) − 12 [e j(e j( f )) − M∇e j e j( f )](01, gradλ2hµ

2 )

so

(4.8)

Trg(∇iy0 )2τ f (iy0 ) = (m∑

j=1( M∇e j

M∇e j −M∇ M∇e j e j )gradg f , 02)

− 12 (01,

m∑j=1

[e j(e j( f )) − 2 M∇e j e j( f )]gradλ2hµ2 )

= (Trg( M∇)2gradg f , 02) − 12 (01, (∆g( f ) −

m∑j=1

M∇e j e j( f )) gradλ2hµ2 ).


On the other hand, by (3.2)

(4.9)

m∑j=1

R((e j, 02), τ f (iy0 )

)(e j, 02)

=m∑

j=1−m

2 R((e j, 02), (0, f · gradλ2hµ

2 ))(e j, 02)

+R((e j, 02), (gradg f , 02)

)(e j, 02)

= − m8λ2 f (gradλ2hµ

2)(µ2)m∑

j=1g(gradµ2gλ

2, e j)(e j, 02)

−mµ2

4λ2

m∑j=1

g( M∇e j gradµ2gλ2 − 1

2λ2 e j(λ2)gradµ2gλ2, e j)(01, f · gradλ2hµ

2 )

−m2 f

4 (01,N∇gradλ2hµ

2 gradλ2hµ2 − 1

2µ2 f · gradλ2hµ2(µ2)gradλ2hµ

2 )

+m2 f8λ2 gradλ2hµ

2(µ2)(gradµ2gλ2, 02 )

+(RieM(gradg f ), 02 ) − m4µ2 |gradλ2hµ

2|2m∑

j=1(gradg f , 02 )

+ 14µ2 |gradλ2hµ

2|2m∑

j=1g(gradg f , e j)(e j, 02 )

= − m8λ2 f |gradλ2hµ

2|2(gradµ2gλ2, 02)

−mµ2

4λ2 f [∆g( f ) − 12λ2

m∑j=1

(e j(λ2))2](01, ·gradλ2hµ2 )

−m2 f

4 (01,N∇gradλ2hµ

2 gradλ2hµ2 − 1

2µ2 f |gradλ2hµ2|2gradλ2hµ

2 )


2|2(gradµ2gλ2, 02 )

+(RieM(gradg f ), 02 ) − m2

4µ2 |gradλ2hµ2|2(gradg f , 02 )

+ 14µ2 |gradλ2hµ

2|2(gradg f , 02 )

thus (4.4) gives

(4.10)

τ2f (iy0 ) = f (Trg( M∇)2gradg f − m


2



− m2


4µ2 |gradλ2hµ2|2gradg f , 02)

−( M∇gradg f gradg f , 02 ) + 12 |gradg f |2(01, gradλ2hµ

2) ),

+ f (01,−12 (∆g( f ) −

n∑α=1

M∇e j e j( f )) gradλ2hµ2

−m2 f



2|2gradλ2hµ2] ).

Therefore, the inclusion map iy : (M, g) → (M ×(µ,λ) N, g),∀y ∈ M is a proper f -harmonicmap (τ2

f (iy0 ) = 0) if and only if

(4.11)f (Trg( M∇)2gradg f − m


2



− m2


4µ2 |gradλ2hµ2|2gradg f − M∇gradg f gradg f = 0

and

(4.12)− 1

2 |gradg f |2gradλ2hµ2 + f (− 1

2 (∆g( f ) − M∇e j e j( f )) gradλ2hµ2

−m2 f



2|2gradλ2hµ2] ) = 0.

For inclusion map ix0 , in the same way, since

(4.13) τ f (ix0 ) = −n2

( f · gradµ2gλ2, 02) + (01, gradh f )

8 WEI-JUN LU

(4.14)

τ2f (ix0 ) = f

n∑α=1[∇

ix0eα ∇

ix0eα − ∇

ix0N∇eα eα

]τ f (ix0 )

+R((eα, 02), τ f ix0

)(eα, 02) − ∇(01,gradh f )dix0 (τ f (ix0 ))

= (01, f Trh( N∇)2gradh f ) − 12 f [∆h( f )

−n∑α=1

N∇eα eα( f )]( gradµ2gλ2, 02)

+ n−18µ2 n f |gradλ2hµ

2|2)(01, gradλ2hµ2 )

− nλ24µ2 f [∆h( f ) − 1

2µ2

n∑α=1

(eα(µ2))2]( gradµ2gλ2, 02)

−n2 f

4 [ M∇gradµ2gλ2 gradµ2gλ

2 − 12λ2 f |gradµ2gλ

2|2gradµ2gλ2, 02 )

+(01, RieN(gradh f ) ) − n2−14λ2 |gradµ2gλ

2|2(01, gradh f )−(01,

N∇gradh f gradh f ) + 12 |gradh f |2(gradµ2gλ

2, 02 )

we obtain

Theorem 4.2. Let f : (N, h) → (0,+∞) be a smooth function. The inclusion map of themanifold (N, h) into the nontrivial (proper) doubly warped product manifold M ×(µ,λ) N isa proper f -biharmonic map if and only if λ, µ, f satisfy

(4.15)f Trh( N∇)2gradh f + n−1

8µ2 n f |gradλ2hµ2|2) gradλ2hµ

2

+RieN(gradh f ) ) − n2−14λ2 |gradµ2gλ

2|2gradh f− N∇gradh f gradh f = 0,

and

(4.16)

−n2 f

4 ( M∇gradµ2gλ2 gradµ2gλ

2 − 12λ2 f |gradµ2gλ

2|2gradµ2gλ2 )

+ 12 |gradh f |2gradµ2gλ

2 − 12 f [∆h( f ) −

n∑α=1

N∇eα eα( f )] gradµ2gλ2

− nλ24µ2 f [∆h( f ) − 1

2µ2

n∑α=1

(eα(µ2))2]gradµ2gλ2 = 0

Remark 4.3. We don’t observe that whether f -tension field τ2f (iy0 ) and τ2

f (ix0 ) have solution.If when f = const. or λ = const. or µ = const. can ensure τ2

f (iy0 ) = 0 or τ2f (ix0 ) = 0, then

we can conclude that there never exist proper f -biharmonic maps for the inclusion case assame as the result in [SE] and [Lu].

5. f -

In this section we shall give a method to construct proper f -biharmonic maps of producttype.

An obvious verification shows that Ψ = IdM × ϕN : (M × N, g ⊕ h) → (M × N, g ⊕ h)is a harmonic map if IdM : (M, g) → M is a identity map and ϕN : (N, h) → N is aharmonic map. However, with somewhat modification, one can replace the product metricg⊕h on M×N (either as domain or codomain) by the doubly warped product metric tensorg = µ2g ⊕ λ2h, then the product map may fail to remain harmonic. For this reason, we canfind out what condition on λ and µ so that the product map preserves its f -biharmonicity.

We first take into account the product map

(5.1) Ψ = IdM × ϕN : (Mm ×(µ,λ) Nn, g)→ (M × N, g ⊕ h),Ψ(x, y) = (x, ϕN(y)).

We obtain


Theorem 5.1. Let (Mm, g) and (Nn, h) be Riemannian manifolds, and let λ ∈ C∞(M),µ ∈ C∞ and f ∈ C∞(M × N) be three positive functions. Suppose that ϕN : N → Nis a harmonic map. Then the product map Ψ = IdM × ϕN defined by (5.1) is a properf -biharmonic map if and only if λ, µ satisfy

(5.2)fµ2 Trg( M∇)2( f · gradµ2g log( fλn)

)+ f n M∇gradµ2g log µ f · gradµ2g log( fλn)

+f 2

µ2 RicM(gradµ2g log( fλn))− M∇gradµ2g f f · gradµ2g log( fλn) = 0

and(5.3)

fλ2 Trh( N∇dϕN )2 f · dϕN

(gradλ2h log( fµm)

)+ f m N∇gradλ2g log µdϕN

(f gradλ2h log( fµm)

+f 2

λ2 TrhRN(dϕN , gradλ2h log( fµm))dϕN −

N∇dϕN (gradλ2h f ) f dϕN(gradλ2h log( fµm)

)= 0.

Proof. Let e jmj=1 be an local orthonormal frame on (M, g) and eαnα=1 be an orthonormal

frame on (N, h). Then ( 1µe j, 02), (01,

1λeα) j=1,...,m,α=1,...,n is an local orthonormal frame on

the doubly warped product manifold Mm ×(µ,λ) Nn. By (3.1), together with the assumptionφN is harmonic, i.e. τ(φN) = 0, we have(5.4)

τ(Ψ) =m∑

j=1

( 1µ2

M∇dΨ(e j,02)dΨ(e j, 02) − 1µ2 dΨ( ∇(e j,02)(e j, 02) )

)+

n∑α=1

( 1λ2

N∇IdT M×dφN (01,eα)IdT M × dφ(01, eα) − 1λ2 IdT M × dφN( ∇(01,eα)(01, eα) )

)= n( gradµ2g log λ, 02 ) + m( 01, dφN(gradλ2h log µ) ) + 1

λ2 (01, τ(φ))= n( gradµ2g log λ, 02 ) + m( 01, dφN(gradλ2h log µ) )

and

(5.5)

τ f (Ψ) = f n( gradµ2g log λ, 02 ) + f m( 01, dφN(gradλ2h log µ) )+IdT M × dφN(gradµ2g f , gradλ2h f )= ( f n gradµ2g log λ + gradµ2g f , 02 )+( 01, f m dφN( gradλ2h log µ) + dφN(gradλ2h f ) )= f ( gradµ2g log fλn, dφN( gradλ2h log fµm) ).

Since from (2.6) the f-biharmonic map of Ψ is

(5.6)

τ2f (Ψ) = f [Trg(∇Ψ)2τ f (Ψ) + TrgR(dΨ, τ f (Ψ) )dΨ] − ∇Ψgradg f τ f (Ψ)

= fm∑

j=1

([∇Ψ

( 1µ e j,02)

∇Ψ( 1µ e j,02)

− ∇Ψ∇( 1µ e j ,02)(

1µ e j,02)

]τ f (Ψ)

+R(( 1µe j, 02), τ f Ψ

)( 1µe j, 02)

+ fn∑α=1

([∇Ψ

(01,1λ eα)∇Ψ

(01,1λ eα)− ∇Ψ

∇(01 ,1λ eα )(01,

1λ eα)

]τ f (Ψ)

+R((01, dφN( 1

λeα) ), τ f (Ψ)

)(01, dφN( 1

λeα) )

−∇Ψ(gradµ2g f , gradλ2h f )τ f (Ψ),

we in the position computer the following terms:

∇Ψ( 1µ e j,02)

τ f (Ψ) = 1µ( M∇e j f · gradµ2g log( fλn), 02),

(5.7)

∇Ψ∇( 1µ e j ,02)(

1µ e j,02)

τ f (Ψ) = 1µ2∇

Ψ

( M∇e j ,02)− 12 (01,gradλ2hµ

2)τ f (Ψ)

= 1µ2 ( M∇ M∇e j e j f · gradµ2g log( fλn), 02 )

−( 01,N∇dϕN (gradλ2hlogµ) f dϕN( gradλ2h log( fµm) ) ),

10 WEI-JUN LU

(5.8)∇Ψ(gradµ2g f , gradλ2h f )τ f (Ψ) = ( M∇gradµ2g f f · gradµ2g log( fλn), 02 )

+( 01,N∇dϕN (gradλ2h f ) f dϕN( gradλ2h log( fµm) ) ),

(5.9)∇Ψ

( 1µ e j,02)

∇Ψ( 1µ e j,02)

τ f (Ψ) = ∇Ψ( 1µ e j,02)

1µ( M∇e j f · gradµ2g log( fλn), 02)

= 1µ2 ( M∇e j

M∇e j f · gradµ2g log( fλn), 02 ),

∇Ψ(01,

1λ eα)τ f (Ψ) = 1

λ∇(01,dϕN (eα)τ f (Ψ) = ( 01,

1λ

N∇dϕN (eα) f dϕN( gradλ2g log( fµm) ),

(5.10)

∇Ψ∇(01 ,

1λ eα )(01,

1λ eα)τ f (Ψ) = 1

λ2∇Ψ

(01, N∇eα eα)− 12 (gradµ2gλ

2,02)τ f (Ψ)

= 1λ2 (01,

N∇dϕN ( N∇eα eα) f dϕN( gradλ2g log( fµm) ) )−( M∇gradµ2g log λ f · gradµ2g log( fλn), 02 ),

(5.11)∇Ψ

(01,1λ eα)∇Ψ

(01,1λ eα)τ f (Ψ) = 1

λ2∇(01, dϕN (eα)( 01,N∇dϕN (eα) f dϕN( gradλ2g log( fµm) )

= 1λ2 ( 01,

N∇dϕN (eα)N∇dϕN (eα) f dϕN( gradλ2g log( fµm) )

From (5.7),(5.9)-(5.11), we have

(5.12)

Trg(∇Ψ)2τ f (Ψ) = ( 1µ2 Trg( M∇2) f · gradµ2g log( fλn)

+n M∇gradµ2g log λ f · gradµ2g log( fλn), 02 )+( 01,

1λ2 Trh( N∇dϕN ))2 f dϕN( gradλ2h log( fµm)

+m N∇dϕN (gradλ2h log µ) f dϕN( gradλ2h log( fµm) ) )

On the other hand, since(5.13)

m∑j=1

R(( 1µe j, 02), τ f (Ψ)

)( 1µe j, 02) = 1

µ2

m∑j=1

(∇(e j,02)∇( f ·gradµ2g log( fλn),02)(e j, 02)

+∇( f ·gradµ2g log( fλn),02)∇(e j,02)(e j, 02)−∇[(e j,02), ( f ·gradµ2g log( fλn),02)(e j, 02)

+R((e j, 02), (01, f dϕN( gradλ2h log( fµm))e j, 02 )(e j, 02))

= 1µ2

( m∑j=1

RM(e j, f · gradµ2g log( fλn))e j, , 02)

=fµ2 (RicM(gradµ2g log( fλn)), 02),

and similarly

(5.14)

n∑α=1

R((01, dφN( 1

λeα) ), τ f (Ψ)

)(01, dφN( 1

λeα) )

= 1λ2 (01,

n∑α=1

RN(dφN(eα), f · gradλ2h log( fµm))dφN(eα) )

=fλ2 ( 01, TrhRN(dφN , f · gradλ2h log( fµm))dφN ),


(5.6) gives

(5.15)

τ2f (Ψ) = ( f

µ2 Trg( M∇2) f · gradµ2g log( fλn) + f 2

µ2 RicM(gradµ2g log( fλn))+ f n M∇gradµ2g log λ f · gradµ2g log( fλn)− M∇gradµ2g f f · gradµ2g log( fλn), 02 )+( 01,

fλ2 Trh( N∇dϕN ))2 f dϕN( gradλ2h log( fµm)

+ f m N∇dϕN (gradλ2h log µ) f dϕN( gradλ2h log( fµm) )+

f 2

λ2 TrhRN(dφN , f · gradλ2h log( fµm))dφN

− N∇dϕN (gradλ2h f ) f dϕN( gradλ2h log( fµm) ) ).

Therefore, by τ2f (Ψ) = 0 we conclude the statement.

In order to find the relation between product map and the projection map, we first in-vestigate the f -bitension field of the projection and give the following lemmas.

Lemma 5.2. Let the projection π1 : (Mm×(µ,λ) Nn, g)→ (Mm, g), π1(x, y) = x, be a doublywarped product onto its first factor and let f : M×N → (0,+∞) be smooth function. Thenthe f -bitension field of π1 is

(5.16)τ2

f (π1) = fµ2 Trg( M∇2) f · gradµ2g log( fλn) + f 2

µ2 RicM(gradµ2g log( fλn))+ f n M∇gradµ2g log λ f · gradµ2g log( fλn) − M∇gradµ2g f f · gradµ2g log( fλn).

Proof. By a similar calculation as (5.4) and (5.5), we have

(5.17)

τ(π1) = Trg∇dπ1 =m∑

j=1

( 1µ2

M∇dπ1(e j,02)dπ1(e j, 02) − 1µ2 dπ1( ∇(e j,02)(e j, 02) )

)+

n∑α=1

( 1λ2

N∇dπ1(01,eα)dπ1(01, eα) − 1λ2 dπ1( ∇(01,eα)(01, eα) )

)= n

2λ2 gradµ2gλ2 | π1

= n gradµ2g log λ | π1,

(5.18)τ f (π1) = f n gradµ2g log λ π1 + dπ1(gradµ2g f , gradλ2h f )

= f gradµ2g log(λn f ) | π1.

Since∇π1

( 1µ e j,02)

τ f (π1) = 1µ( M∇dπ1(e j,0)dπ1(τ f (π1)) = 1

µM∇e j f · gradµ2g log( fλn),

(5.19)∇π1

∇( 1µ e j ,02)(

1µ e j,02)

τ f (π1) = 1µ2∇

π1

( M∇e j ,02)− 12 (01,gradλ2hµ

2)τ f (π1)

= 1µ2

M∇ M∇e j e j f · gradµ2g log( fλn),

(5.20) ∇π1(gradµ2g f , gradλ2h f )τ f (π1) = M∇gradµ2g f f · gradµ2g log( fλn),

(5.21)∇π1

( 1µ e j,02)

∇π1

( 1µ e j,02)

τ f (π1) = ∇π1

( 1µ e j,02)

1µ

M∇e j f · gradµ2g log( fλn)

= 1µ2

M∇e jM∇e j f · gradµ2g log( fλn),

∇π1

(01,1λ eα)τ f (π1) = M∇0dπ1(τ f (π1) = 0,

(5.22)∇π1

∇(01 ,1λ eα )(01,

1λ eα)τ f (π1) = 1

λ2∇π1

(01, N∇eα eα)− 12 (gradµ2gλ

2,02)τ f (π1)

= − M∇gradµ2g log λ) f · gradµ2g log( fλn),

12 WEI-JUN LU

(5.23) ∇π1

(01,1λ eα)∇π1

(01,1λ eα)τ f (π1) = 0,

from (5.19),(5.21)-(5.23), we have

(5.24)Trg(∇π1 )2τ f (π1) = 1

µ2 Trg( M∇2) f · gradµ2g log( fλn)+n M∇gradµ2g log λ f · gradµ2g log( fλn).

On the other hand, since

(5.25)

TrgRM(dπ1, τ f (π1))dπ1 =m∑

j=1RM(dπ1( 1

µe j, 02), τ f (π1)

)dπ1( 1

µe j, 02)

+n∑α=1

RM(dπ1(01,1λeα), τ f (π1)

)dπ1(01,

1λeα )

=fµ2

m∑j=1

RM(e j, gradµ2g log( fλn))e j

=fµ2 RicM(gradµ2g log( fλn)),

we get

(5.26)

τ2f (π1) = f

(Trg(∇π1 )2τ f (π1) + TrgRM(dπ1, τ f (π1))dπ1

)− ∇

π1gradg f τ f (π1)

=fµ2 Trg( M∇2) f · gradµ2g log( fλn) + f 2

µ2 RicM(gradµ2g log( fλn))+ f n M∇gradµ2g log λ f · gradµ2g log( fλn) − M∇gradµ2g f f · gradµ2g log( fλn).

Thus we end the proof.

By a similar discussion for the projection π2 : M ×(µ,λ) N → N, we have

Lemma 5.3. Let the projection π2 : (Mm ×(µ,λ) Nn, g)→ (Nn, h), π1(x, y) = y, be a doublywarped product onto its second factor and let f : M × N → (0,+∞) be smooth function.Then the f -bitension field of π2 is

(5.27)τ2

f (π2) =fλ2 Trh

N∇2 f · gradλ2h log( fµm) + f m N∇gradλ2g log µ f gradλ2h log( fµm)

+f 2

λ2 RieN(gradλ2h log( fµm))− N∇gradλ2h f f · gradλ2h log( fµm).

Observe the f -bitensions in Theorem 5.1 and Lemma 5.2, we easily obtain

Corollary 5.4. The product map Ψ : Mm ×(µ,λ) Nn → M × N,Ψ(x, y) = (x, ϕN(y)) withharmonic map ϕ is a proper f -harmonic map with f ∈ C∞(M × N) if and only if theprojections π1 is a proper f -biharmonic map and

(5.28)

fλ2 Trh( N∇dϕN )2 f · dϕN

(gradλ2h log( fµm)

)+ f m N∇gradλ2g log µdϕN

(f gradλ2h log( fµm) + f 2

λ2 TrhRN(dϕN , gradλ2h log( fµm))dϕN

− N∇dϕN (gradλ2h f ) f dϕN(gradλ2h log( fµm)

)= 0.

Now we proceed to treat the product map Ψ = ˜ϕM × IdN :

(5.29) (Mm ×(µ,λ) Nn, g = µ2g ⊕ λ2h)→ (M × N, g ⊕ h), Ψ(x, y) = (ϕM(x), y).

of the harmonic map ϕM : M → M and the identity map IdN : N → N. Similar method asTheorem 5.1, together with Lemma 5.3, we have


Theorem 5.5. With the same notions of λ, µ, f as stated in Theorem 5.1. The product mapΨ = ˜ϕM × IdN defined by (5.29) is a proper f -harmonic map if and only λ, µ satisfy

(5.30)

fµ2 Trh( M∇dϕM )2 f · dϕM

(gradµ2g log( fλn)

)+ f n M∇dϕM (gradµ2g log µ)dϕM

(f · gradµ2g log( fλn)

)+

f 2

µ2 TrgRM(dϕM , gradµ2g log( fλn))dϕM

− M∇dϕM (gradµ2g f ) f dϕM(gradµ2g log( fλn)

)= 0

and the projections π2 is a proper f -biharmonic map, that is

(5.31)τ2

f (π2) =fλ2 Trh

N∇2 f · gradλ2h log( fµm) + f m N∇gradλ2g log µ f gradλ2h log( fµm)

+f 2

λ2 RieN(gradλ2h log( fµm))− N∇gradλ2h f f · gradλ2h log( fµm) = 0.

In the remainder of this section, we turn to consider the case of the product map

(5.32) Ψ = IdM × ϕN : (M × N, g ⊕ h)→ (Mm ×(µ,λ) Nn, g), Ψ(x, y) = (x, ϕN(y));

that is the case of the product metric on the codomain is doubly warped metric. We willsee that the harmonic energy density e(ϕN) has an important role for the f -harmonicity ofthe product map Ψ = IdM × ϕN .

Let (e j, 02), (01, eα), j = 1, . . . ,m, α = 1, . . . , n is an local orthonormal frame on theusual product manifold Mm×Nn, where e j

mj=1 and eαnα=1 respectively denote orthonormal

frames on (M, g) and on (N, h). By τ(ϕN) = 0, the tension field of Ψ is

(5.33)

τ(Ψ) = Trg⊕h∇dΨ =m∑

j=1

(∇dΨ(e j,02)dΨ(e j, 02) − dΨ(∇(e j,02)(e j, 02) )

)+

n∑α=1

(∇IdT M×dϕN (01,eα)IdT M × dϕN(01, eα)

−IdT M × dϕN(∇(01,eα)(01, eα) ))

= −m2 (01, gradλ2hµ

2) + (01, τ(ϕN) ) − 12

n∑α=1ϕ∗Nh(eα, eα)(gradµ2gλ

2, 02)

= (01,−m2 gradλ2hµ

2) − ( e(ϕN)gradµ2gλ2, 02),

thus we get

(5.34)

τ f (Ψ) = (01,−m2 f gradλ2hµ

2) + (− e(ϕN) f gradµ2gλ2, 02)

+IdT M × dϕN(gradg f , gradh f )= (01,−

m2 f gradλ2hµ

2 + dϕN(gradh f ) )+(− e(ϕN) f gradµ2gλ

2 + gradg f , 02 ).

Since from (2.6) the f-biharmonic map of Ψ is

(5.35)

τ2f (Ψ) = f [Trg⊕h(∇Ψ)2τ f (Ψ) + Trg⊕hR(dΨ, τ f (Ψ) )dΨ] − ∇Ψgradg⊕h f τ f (Ψ)

= fm∑

j=1

([∇Ψ(e j,02)∇

Ψ(e j,02) − ∇

Ψ(∇e j ,02)(e j,02)]τ f (Ψ)

+R((e j, 02), τ f Ψ

)(e j, 02)

)+ f

n∑α=1

([∇Ψ(01,eα)∇

Ψ(01,eα) − ∇

Ψ∇(01 ,eα )(01,eα)]τ f (Ψ)

+R((01, dφN(eα) ), τ f (Ψ)

)(01, dφN(eα)

)−∇Ψ(gradg f , gradh f )τ f (Ψ),

14 WEI-JUN LU


∇Ψ(e j,02)τ f (Ψ) = −m2∇(e j,02)(01, f dϕN(gradλ2hµ

2) ) + ∇(e j,02)(01, dϕN(gradh f )−∇(e j,02)(e(ϕN) f gradµ2gλ

2, 02 ) + ∇(e j,01)(gradg f , 02 )= −

m f4λ2 e j(λ2)(01, dϕN(gradλ2hµ

2 ) − m f4µ2

(dϕN(gradλ2hµ

2)(µ2))(e j, 02)

+ 12λ2 e j(λ2)(01, dϕN(gradh f ) + 1

2µ2

(dϕN(gradh f )(µ2)

)(e j, 02)

−( M∇e j e(ϕN) f gradµ2gλ2, 02 ) + 1

2 g(e j, e(ϕN) f gradµ2gλ2)(01, gradλ2hµ

2) )+( M∇e j gradg f , 02) − 1

2 g(e j, gradg f )(01, gradλ2hµ2),

∇Ψ∇(e j ,02)(e j,02)(τ f (Ψ)) = ∇Ψ

( M∇e j e j,02)τ f (Ψ)

= −m f4λ2

( M∇e j e j(λ2))(01, dϕN(gradλ2hµ

2) ) −m f4µ2

(dϕN(gradλ2hµ

2)(µ2))( M∇e j e j, 02)

+1

2λ2M∇e j e j(λ2)(01, dϕN(gradh f ) ) +

12µ2


)( M∇e j e j, 02)

− ( M∇ M∇e j e j e(ϕN) f gradµ2gλ2, 02 )

+12

g( M∇e j e j, e(ϕN) f gradµ2gλ2)(01, gradλ2hµ

2) )

+ ( M∇ M∇e j e j gradg f , 02) −12

g( M∇e j e j, gradg f )(01, gradλ2hµ2),

∇Ψ(gradg f , gradh f )(τ f (Ψ)) =(∇(gradg f ,02) + ∇(01,dϕN (gradh f )

)dΨ(τ f (Ψ))

= −m f4λ2

(gradg f (λ2)

)(01, dϕN(gradλ2hµ

2) ) −m f4µ2

(dϕN(gradλ2hµ

2)(µ2))(gradg f , 02)

+1

2λ2

(gradg f (λ2)

)(01, dϕN(gradh f ) ) − ( M∇gradg f e(ϕN) f gradµ2gλ

2, 02 )

+1

2µ2


)(gradg f , 02)

+12

g(gradg f , e(ϕN) f gradµ2gλ2)(01, gradλ2hµ

2 )

+ ( M∇gradg f gradg f , 02) −12|gradg f |2(01, gradλ2hµ

2)

−m2

(01,N∇dϕN (gradh f ) f dϕN(gradλ2hµ

2) )

−m f4

h(dϕN(gradh f ), dϕN(gradλ2hµ2) )(gradµ2gλ

2, 02)

+ (01,N∇dϕN (gradh f )dϕN(gradh f ) )

−12|dϕN(gradh f )|2(gradµ2gλ

2, 02)

−1

2µ2 e(ϕN) f(dϕN(gradh f )(µ2)

)(gradµ2gλ

2, 02 )

−1

2λ2 e(ϕN) f |gradµ2gλ2|2(01, dϕN(gradh f ) )

+1

2µ2


)(gradg f , 02 ) +

12λ2

(gradh f (λ2)

)(01, dϕN(gradh f ) )

(5.36)


∇Ψ(e j,02)∇Ψ(e j,02)τ f (Ψ) = −

m f8λ4 (e j(λ2))2(01, dϕN(gradλ2hµ

2) )

−m f

8λ2µ2 e j(λ2)(dϕN(gradλ2hµ2))(µ2)(e j, 02)

−m

4µ2 ( M∇e j

(dϕN(gradλ2hµ

2)(µ2))e j, 02 ) +

m f8µ2

(dϕN(gradλ2hµ

2)(µ2))(01, gradλ2hµ

2 )

+1

4λ4 (e j(λ2))2(01, dϕN(gradh f ) ) +1

4λ2µ2 e j(λ2)(dϕN(gradh f )(µ2)

)(e j, 02)

+1

2µ2 ( M∇e j


)e j, 02 ) −

14µ2


)(01, gradλ2hµ

2 )

− ( M∇e jM∇e j e(ϕN) f gradµ2gλ

2, 02 ) +12

g(e j,M∇e j e(ϕN) f gradµ2gλ

2)(01, gradλ2hµ2)

+1

4λ2 e j(λ2)g(e j, e(ϕN) f gradµ2gλ2)(0, gradλ2hµ

2 )

+1

4µ2 g(e j, e(ϕN) f gradµ2gλ2)|gradλ2hµ

2|2(e j, 02 )

+ ( M∇e jM∇e j gradg f , 02 ) −

12

g(e j,M∇e j gradg f )(01, gradλ2hµ

2)

−1

4λ2 e j(λ2)g(e j, gradg f )(01, gradλ2hµ2 ) −

14µ2 g(e j, gradg f )|gradλ2hµ

2|2(e j, 02),

(5.37)

∇Ψ(01,eα)τ f (Ψ) = ∇(01,dϕN (eα) )dΨ(τ f (Ψ))

= −m2

(01,N∇dϕN (eα) f dϕN(gradλ2hµ

2) )

+m f4

h(dϕN(eα), dϕN(gradλ2hµ2))(gradµ2gλ

2, 02)

+ (01,N∇dϕN (eα)dϕN(gradh f ) −

12

h(dϕN(eα), dϕN(gradh f ))(gradµ2gλ2, 02)

−1

2λ2 e(ϕN) f |gradµ2gλ2|2(01, dϕN(eα) ) −

12µ2 e(ϕN) f

(dϕN(eα)(µ2)

)(gradµ2gλ

2, 02)

+1

2λ2

((gradg f )(λ2)

)(01, dϕN(eα) ) +

12µ2

(dϕN(eα)(µ2)

)(gradg f , 02),

∇Ψ∇(01 ,eα )(01,eα)τ f (Ψ) = −m2

(01,N∇dϕN ( N∇eα eα) f dϕN(gradλ2hµ

2) )

+m f4

h(dϕN( N∇eα eα), dϕN(gradλ2hµ2))(gradµ2gλ

2, 02)

+ (01,N∇dϕN ( N∇eα eα)dϕN(gradh f ) −

12

h(dϕN( N∇eα eα), dϕN(gradh f ))(gradµ2gλ2, 02)

−1

2λ2 e(ϕN) f |gradµ2gλ2|2(01, dϕN( N∇eα eα) ) −

12µ2

(dϕN( N∇eα eα)(µ

2))(gradµ2gλ

2, 02)

+1

2λ2

(gradg f (λ2)

)(01, dϕN( N∇eα eα) ) +

12µ2

(dϕN( N∇eα eα)(µ

2))(gradg f , 02),

(5.38)

16 WEI-JUN LU

∇Ψ(01,eα)∇Ψ(01,eα)τ f (Ψ)

= −m2

(01,N∇dϕN (eα)

N∇dϕN (eα) f dϕN(gradλ2hµ2) )

+m f4

h(dϕN(eα), N∇dϕN (eα)dϕN(gradλ2hµ2))(gradµ2gλ

2, 02)

+m f8λ2 h(dϕN(eα), dϕN(gradλ2hµ

2))|gradµ2gλ2|2(01, dϕN(eα))

+m f8µ2

(dϕN(eα)(µ2)

)h(dϕN(eα), dϕN(gradλ2hµ

2))(gradµ2gλ2, 02)

+ (01,N∇dϕN (eα)

N∇dϕN (eα)dϕN(gradh f ) )

−12

h(dϕN(eα), N∇dϕN (eα)dϕN(gradh f )(gradµ2gλ2, 02)

−1

4λ2 h(dϕN(eα), dϕN(gradh f ))|gradµ2gλ2|2(01, dϕN(eα))

−1

4µ2

(dϕN(eα)(µ2)

)h(dϕN(eα), dϕN(gradh f ))(gradµ2gλ

2, 02)

− (01,N∇dϕN (eα)

12λ2 e(ϕN) f |gradµ2gλ

2|2dϕN(eα) )

+1

4λ2

(e(ϕN)

)2 f |gradµ2gλ2|2(gradµ2gλ

2, 02) )

−1

4µ4 e(ϕN) f |dϕN(eα)(µ2)|2(gradµ2gλ2, 02)

−1

4λ2µ2 e(ϕN) f(dϕN(eα)(µ2)

)|gradµ2gλ

2|2(01, dϕN(eα))

+ (01,N∇dϕN (eα)

12λ2

(gradg f (λ2)

)dϕN(eα), )

−1

4λ2

(gradg f (λ2)

)e(ϕ)(gradµ2gλ

2, 02 )

+1

4µ4 |dϕN(eα))(µ2)|2(gradg f , 02)

+1

4λ2µ2

(dϕN(eα)(µ2)

)(gradg f (λ2)

)(01, dϕN(eα) ),

(5.39)

From (5.36),(5.37)-(5.39), we get the complex expression of Trg⊕h(∇Ψ)2τ f (Ψ).On the other hand, since

m∑j=1

R((e j, 02), τ f (Ψ)

)(e j, 02)

=

m∑j=1

R((e j, 02), (01, f dϕN(gradλ2hµ

2) ))(e j, 02) + R

((e j, 02), (0, dϕN(gradh f ) )

)(e j, 02)

+ R((e j, 02), (e(ϕN) f gradµ2gλ

2, 02 ))(e j, 02) + R

((e j, 02), (gradg f , 02)

)(e j, 02)

(5.40)


=1

4λ2 f(dϕN(gradλ2hµ

2)(µ2)) m∑

j=1

g(gradµ2gλ2, e j)(e j, 02)

+µ2

2λ2

m∑j=1

g( M∇e j gradµ2gλ2 −

12λ2 e j(λ2)gradµ2gλ

2, e j)(01, f dϕN(gradλ2hµ2) )

+m f2

(01,N∇dϕN (gradλ2hµ

2)gradλ2hµ2 −

12µ2 f

(dϕN(gradλ2hµ

2)(µ2))gradλ2hµ

2 )

−m f4λ2 dϕN(gradλ2hµ

2)(µ2)(gradµ2gλ2, 02 )

+1

4λ2 (dϕN(gradh f )(µ2)m∑

j=1

g(gradµ2gλ2, e j)(e j, 02)

+µ2

2λ2

m∑j=1

g( M∇e j gradµ2gλ2 −

12λ2 e j(λ2)gradµ2gλ

2, e j)(01, dϕN(gradh f ) )

+m2

(01,N∇dϕN (gradh f )gradλ2hµ

2 −1

2µ2 dϕN(gradh f )(µ2)gradλ2hµ2 )

−m

4λ2 dϕN(gradh f )(µ2)(gradµ2gλ2, 02 )

+ (RieM(e(ϕN) f gradµ2gλ2), 02 ) −

m4µ2 e(ϕN) f |gradλ2hµ

2|2(gradµ2gλ2, 02 )

+1

4µ2 e(ϕN) f |gradλ2hµ2|2

m∑j=1

g(gradµ2gλ2, e j)(e j, 02 )

+ (RieM(gradg f ), 02 ) −m

4µ2 |gradλ2hµ2|2(gradg f , 02 )

+1

4µ2 |gradλ2hµ2|2

m∑j=1

g(gradg f , e j)(e j, 02 ),

18 WEI-JUN LU

and similarly

n∑α=1

R((01, dϕN(eα) ), τ f (Ψ)

)(01, dϕN(eα) )

=

n∑α=1

R((01, dϕN(eα) ), (01, f dϕN(gradλ2hµ

2) ))(01, dϕN(eα) )

+ R((01, dϕN(eα) ), (01, dϕN(gradh f ) )

)(01, dϕN(eα) )

+ R((01, dϕN(eα) ), (e(ϕN) f gradµ2gλ

2, 02 ))(01, dϕN(eα) )

+ R((01, dϕN(eα) ), (gradg f , 02)

)(01, dϕN(eα) )

= (01,Trh(RN(dϕN , dϕN(gradh f )dϕN ) −1

2λ2 e(ϕN)|gradµ2gλ2|2(01, dϕN(gradh f ) )

+1

4λ2 |gradµ2gλ2|2

n∑α=1

h(dϕN(gradh f , dϕN(eα))(01, dϕN(eα) )

+ (01,Trh(RN(dϕN , dϕN(gradh f )dϕN ) −1

2λ2 e(ϕN) f |gradµ2gλ2|2(01, dϕN(gradh f ) )

+1

4λ2 f |gradµ2gλ2|2

n∑α=1

h(dϕN(gradh f , dϕN(eα))(01, dϕN(eα) )

+λ2

2µ2

n∑α=1

h( N∇dϕN (eα)gradλ2hµ2

−1

2µ2 e(ϕN) f h( dϕN(eα))(µ2)gradλ2hµ2, dϕN(eα) )( e(ϕN) f gradµ2gλ

2, 02 )

+1

4µ2 e(ϕN) f | gradµ2gλ2|2

n∑α=1

h(gradλ2hµ2, dϕN(eα) )(01, dϕN(eα) )

+ (e(ϕN))2 f g( M∇gradµ2gλ2 gradµ2gλ

2 −1

2λ2 |gradµ2gλ2|2gradµ2gλ

2, 02 )

−1

2µ2 (e(ϕN))2 f |gradµ2gλ2|2(01, gradλ2hµ

2) )

(5.41)

+λ2

2µ2

n∑α=1

h( N∇dϕN (eα)gradλ2hµ2

−1

2µ2 e(ϕN) f h( dϕN(eα))(µ2)gradλ2hµ2, dϕN(eα) )( gradg f , 02 )

+1

4µ2 (gradg f )(λ2)n∑α=1

h(gradλ2hµ2, dϕN(eα) )(01, dϕN(eα) )

+ e(ϕN)( M∇gradg f gradµ2gλ2 −

12λ2 (gradg f )(λ2)gradµ2gλ

2, 02 )

−1

2µ2 e(ϕN)(gradg f )(λ2)(01, gradλ2hµ2) ),


noting thatm∑

j=1

g(gradµ2gλ2, e j)(e j, 02) =

m∑j=1

e j(λ2)(e j, 02) = (gradgλ2, 02)£

n∑α=1

h(dϕN(gradh f , dϕN(eα))(01, dϕN(eα) ) = (01, dϕN(gradh f )??

thus (5.35) gives

τ2f (Ψ) = (A, 02) + (01, B)(5.42)

However, the terms ”A” and ”B” in (5.42) all include many complex terms.

Remark 5.6. Unfortunately, we cannot further simplified τ2f , so we don’t obtain very inter-

esting result when τ2f = 0. In fact, even though the case for biharmonic map or f -harmonic

map, we still have nothing good results, see [SE] and [Lu].

6. f -

In this subsection, we consider the general product case, i.e., we modify the identity mapIdM to a harmonic map ϕM : (M, g) → M in Theorem 5.1, then we obtain the followingresult.

Theorem 6.1. Let (Mm, g) and (Nn, h) be Riemannian manifolds, and let λ ∈ C∞(M),µ ∈ C∞ and f ∈ C∞(M × N) be three positive functions. Suppose that ϕM : (M, g) →M, ϕN : N → N are two harmonic maps. Then the product map Φ = ϕM × ϕN : (Mm ×(µ,λ)

Nn, g)→ (M × N, g ⊕ h) defined by Φ(x, y) = (ϕM(x), ϕN(y)) is a proper f -harmonic mapif and only if λ and µ satisfy

(6.1)

fµ2 Trg( M∇dφM )2 f dφM(gradµ2g log( fλn) )

+f 2

µ2 TrgRM(dϕM , dϕM(gradµ2g log( fλn)) )dϕM

+ f n M∇dϕM (gradµ2g log λ) f · dϕM(gradµ2g log( fλn) )− M∇gradµ2g f f · gradµ2g log( fλn) = 0

and

(6.2)

fλ2 Trh( N∇dϕN ))2 f dϕN( gradλ2h log( fµm)+ f m N∇dϕN (gradλ2h log µ) f dϕN( gradλ2h log( fµm) )

+f 2


− N∇dϕN (gradλ2h f ) f dϕN( gradλ2h log( fµm) ) = 0.

Proof. By a similar calculation as (5.33) and (5.34), together with the assumption τ(ϕM) =τ(ϕN) = 0, we have(6.3)

τ(Φ) =m∑

j=1

( 1µ2∇dΦ(e j,02)dΦ(e j, 02) − 1

µ2 dΦ( ∇(e j,02)(e j, 02) ))

+n∑α=1

( 1λ2∇dϕM×dϕN (01,eα)dϕM × dϕN(01, eα) − 1

λ2 dϕM × dϕN( ∇(01,eα)(01, eα) ))

= 1µ2 (dϕM(τ(ϕM)), 02) + n(dϕM(gradµ2g log λ), 02 )

+m( 01, dϕN(gradλ2h log µ) ) + 1λ2 (01, dϕN(τ(ϕN)))

= n(dϕM(gradµ2g log λ), 02 ) + m( 01, dϕN(gradλ2h log µ) )

20 WEI-JUN LU

and

(6.4)

τ f (Φ) = f n(dϕM(gradµ2g log λ), 02 ) + f m( 01, dϕN(gradλ2h log µ) )+dϕM × dϕN(gradµ2g f , gradλ2h f )= ( f ndϕM(gradµ2g log λ) + dϕM(gradµ2g f ), 02 )+( 01, f m dϕN( gradλ2h log µ) + dϕN(gradλ2h f ) )= ( f dϕM(gradµ2g log fλn), 02) + (01, f dϕN( gradλ2h log fµm) ).

Since from (6.4) the f-biharmonic map of Φ is

(6.5)

τ2f (Φ) = f [Trg(∇Φ)2τ f (Φ) + TrgR(dΦ, τ f (Φ) )dΦ] − ∇Φgradg f τ f (Φ)

=fµ2

m∑j=1

([∇Φ(e j,02)∇

Φ(e j,02) − ∇

Φ

∇(e j ,02)(e j,02)]τ f (Φ)

+R((dϕM(e j), 02), τ f Φ

)(dϕM(e j), 02)

+fλ2

n∑α=1

([∇Φ(01,eα)∇

Φ(01,eα) − ∇

Φ

∇(01 ,eα )(01,eα)]τ f (Φ)

+R((01, dφN(eα) ), τ f (Φ)

)(01, dφN(eα) )

−∇Φ(gradµ2g f , gradλ2h f )τ f (Φ),


∇Φ(e j,02)τ f (Φ) = ( M∇dϕM (e j) f · dϕM(gradµ2g log( fλn) ), 02),

(6.6)

∇Φ∇(e j ,02)(e j,02)

τ f (Φ) = ∇Φ( M∇e j ,02)− 1

2 (01,gradλ2hµ2)τ f (Φ)

= ( M∇dϕM ( M∇e j e j) f · dϕM(gradµ2g log( fλn) ), 02 )−( 01, µ

2 N∇dϕN (gradλ2hlogµ) f dϕN( gradλ2h log( fµm) ) ),

(6.7)∇Φ(gradµ2g f , gradλ2h f )τ f (Φ) = ( M∇dϕM (gradµ2g f ) f · dϕM(gradµ2g log( fλn) ), 02 )

+( 01,N∇dϕN (gradλ2h f ) f dϕN( gradλ2h log( fµm) ) ),

(6.8)∇Φ(e j,02)∇

Φ(e j,02)τ f (Φ) = ∇Φ(e j,02)(

M∇dϕM (e j) f · dϕM(gradµ2g log( fλn) ), 02)= ( M∇dϕM (e j)

M∇dϕM (e j) f · dϕM(gradµ2g log( fλn) ), 02 ),

∇Φeα)τ f (Φ) = ∇(01,dϕN (eα)τ f (Φ) = ( 01,N∇dϕN (eα) f dϕN( gradλ2g log( fµm) ),

(6.9)∇Φ∇(01 ,eα )(01,eα)

τ f (Φ) = ∇Φ(01, N∇eα eα)− 1

2 (gradµ2gλ2,02)τ f (Φ)

= (01,N∇dϕN ( N∇eα eα) f dϕN( gradλ2g log( fµm) ) )

−λ2( M∇gradµ2g log λ f · gradµ2g log( fλn), 02 ),

(6.10)∇Φ(01,eα)∇

Φ(01,eα)τ f (Φ) = ∇(01, dϕN (eα)( 01,

N∇dϕN (eα) f dϕN( gradλ2g log( fµm) )= ( 01,

N∇dϕN (eα)N∇dϕN (eα) f dϕN( gradλ2g log( fµm) )

From (6.6),(6.8)-(6.10), we have

(6.11)

Trg(∇Φ)2τ f (Φ) = (Trg( M∇dϕM)2 f · dϕMgradµ2g log( fλn)+µ2n M∇dϕM (gradµ2g log λ) f · dϕM(gradµ2g log( fλn) ), 02 )

+Trh( N∇dϕN ))2 f dϕN( gradλ2h log( fµm)+λ2m N∇dϕN (gradλ2h log µ) f dϕN( gradλ2h log( fµm) ) )


On the other hand, since

(6.12)

m∑j=1

R(dϕM(e j), 02), τ f (Φ)

)(dϕM(e j), 02)

=m∑

j=1

(∇(dϕM (e j),02)∇( f ·gradµ2g log( fλn),02)(dϕM(e j), 02)

−∇( f ·gradµ2g log( fλn),02)∇(dϕM (e j),02)(dϕM(e j), 02)−∇[(dϕM (e j),02), ( f ·gradµ2g log( fλn),02)](dϕM(e j), 02)

+R((dϕM(e j), 02), (01, f dϕN( gradλ2h log( fµm))dϕM(e j) )(dϕM(e j), 02))

=( m∑

j=1RM(dϕM(e j), f · gradµ2g log( fλn))dϕM(e j), , 02)

= f (TrgRM(dϕM , dϕM(gradµ2g log( fλn)) )dϕM , 02),

and similarly

(6.13)

n∑α=1

R((01, dφN(eα) ), τ f (Φ)

)(01, dφN(eα) )

= (01,n∑α=1

RN(dφN(eα), f · gradλ2h log( fµm))dφN(eα) )

= f ( 01, TrhRN(dφN , f · gradλ2h log( fµm))dφN ),

(6.6) gives

(6.14)

τ2f (Φ) = ( f

µ2 Trg( M∇dφM )2 f dφM(gradµ2g log( fλn) )

+f 2

µ2 TrgRM(dϕM , dϕM(gradµ2g log( fλn)) )dϕM

+ f n M∇dϕM (gradµ2g log λ) f · dϕM(gradµ2g log( fλn) )− M∇gradµ2g f f · gradµ2g log( fλn), 02 )+( 01,

fλ2 Trh( N∇dϕN ))2 f dϕN( gradλ2h log( fµm)

+ f m N∇dϕN (gradλ2h log µ) f dϕN( gradλ2h log( fµm) )+

f 2


− N∇dϕN (gradλ2h f ) f dϕN( gradλ2h log( fµm) ) ).

Therefore, by τ2f (Φ) = 0 we conclude the statement.

By using the f -tension fields (5.16) and (5.27) of the projections π1 and π2, (6.14) hasthe expression

τ f (Φ) = f (dϕM(τ f (π1)), dϕN(τ f (π2))).So we have

Corollary 6.2. The product map Φ = ϕM × ϕN : Mm ×(µ,λ) Nn → M × N,Φ(x, y) =(ϕM(x), ϕN(y)) with harmonic maps ϕM and ϕN is a proper f -harmonic map if the projec-tions π1 and π2 are all proper f -harmonic map.

R

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22 WEI-JUN LU

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