the energy density gap of harmonic maps between finsler … · 2019. 7. 31. · isrn geometry 5 3....

International Scholarly Research NetworkISRN GeometryVolume 2011, Article ID 374672, 12 pagesdoi:10.5402/2011/374672

Research ArticleThe Energy Density Gap of Harmonic Mapsbetween Finsler Manifolds

Jingwei Han, Yao-yong Yu, and Jing Yu

School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

Correspondence should be addressed to Jingwei Han, [email protected]

Received 13 April 2011; Accepted 24 May 2011

Academic Editors: D. Danielli and S. Kar

Copyright q 2011 Jingwei Han et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We study the energy density function of nondegenerate smooth maps with vanishing tension fieldbetween two real Finsler manifolds. Firstly, we get a variation formula of energy density functionby using moving frame. With this formula, we obtain a rigidity theorem of nondegenerate mapwith vanishing tension field from the Finsler manifold to the Berwald manifold.

1. Introduction

Finsler manifolds are differential manifolds with Finsler metrics. Finsler metrics areRiemannian metrics but without quadratic restriction, which were firstly introduced by B.Riemann in 1854. Harmonic maps are important and interesting in both differential geometryand mathematical physics. Riemannian manifolds and Finsler manifolds are all metric-measure spaces, so we can study the harmonic map between Finsler manifolds by the theoryof harmonic maps on general metric-measure spaces.

By using the volume measure induced from the projective sphere bundle, harmonicmaps between real Finsler manifolds were introduced and investigated in [1–5]. Recently,the author and Shen have studied the harmonic maps on complex Finsler manifolds [6]. In[3] Mo considered the energy functional and the Euler-Lagrange operator of a smooth mapfrom a real Finsler manifold to a Riemannian manifold. In [5], Shen and Zhang give thetension field of the harmonic maps between Finsler manifolds. Recently, Shen and He [1]have simplified the tension field.

Under what conditions of the energy function a harmonic map is a constant mappingor totally geodesic mapping? This is an important and interesting issue in the study ofharmonic maps, which is referred to as the rigidity theorem and studied by many people onthe Riemannian manifold [7, 8]. In [1, 2], Shen and He have obtained some rigidity theorems.In this paper, we get some rigidity theorems for the nondegenerate map with vanishing

2 ISRN Geometry

tension field from the Finsler manifold to the Berwald manifold, which generalize the resultsin [2].

Precisely, we prove the following Bochner-type formula.

Theorem 1.1. Let (M,F) be a Finsler manifold, and let (˜M, ˜F) be a Berwald manifold. If the tensionfield of φ : (M,F) → (˜M, ˜F) is zero, then

∑

j

Sj|j =∣

∣∇dφ∣

∣

2 +∑

i,j

⟨

φ∗ei, φ∗ej⟩bRi

j

−∑

i,j

(

∣

∣φ∗ei∣

∣

2∣∣φ∗ej

∣

∣

2 − ⟨

φ∗ei, φ∗ej⟩⟨

φ∗ei, φ∗ej⟩

)

K˜M(

φ∗ei ∧ φ∗ej)

,

(1.1)

where S =∑

α,i,j φαibφα

i|jωj := Sjω

j . In particular, if M and ˜M are Riemannian manifolds, then∑

j Sj|j is Δe(φ).

Moreover, by using the formula we also prove the following rigidity theorem.

Theorem 1.2. Let M be a compact Finsler manifold of dimension n, and let ˜M be a Berwald manifoldof dimension m. Suppose a, b are positive constants, for any X ∈ π∗TM, bRic(X) ≥ a, and K

˜M ≤ b,whereK˜M is the directional section curvature of ˜M. Suppose the tension field of φ : M → ˜M is zeroand φ is nondegenerate. If

e(

φ) ≤ n

2(n − 1)a

b, (1.2)

then φ is a constant map or totally geodesic map. In particular, if e(φ) ≤ a/2b, then φ must be aconstant map.

Some technical terms above will be explained below. The contents of the paper arearranged as follows. In Section 2, some fundamental definitions and formulas which arenecessary for the present paper are given. In Section 3, we consider the map between Finslermanifolds and get a pull-back formula. In Section 4, a Bochner-type formula from the Finslermanifold to the Berwald manifold is shown. Finally, by using the Bochner type formula, weobtain a rigidity theorem.

2. Finsler Manifold

Let M be an n-dimension smooth manifold, and let π : TM → M be the natural projection.A Finsler metric on M is a function F : TM → [0,∞) satisfying the following properties:

(i) F is smooth on TM \ {0};(ii) F(x, λy) = λF(x, y) for all λ > 0;

(iii) the induced quadratic form g is positively definite, where

g = gij(

x, y)

dxi ⊗ dxj , gij =12

(

F2)

yiyj. (2.1)

ISRN Geometry 3

Here and from now on, [F]yi , [F]yiyj denote ∂F/∂yi, ∂2F/∂yi∂yj , and so forth, and wewill use the following convention of index range unless otherwise stated:

1 ≤ i, j, k, . . . ≤ n, 1 ≤ a, b, c, . . . ≤ n − 1, a = n + a, n = dimM;

1 ≤ α, β, γ · · · ≤ m, m = dim ˜M.(2.2)

The canonical projection π : TM → M gives to a covector bundle π∗T ∗M whichhas a global section ω = [F]yidxi called the Hilbert form, whose dual vector field is l =(yi/F)(∂/∂xi) = li(∂/∂xi), viewed as a global section of the pull-back bundle π∗TM. Wehave the following important quantities:

A := Aijkdxi ⊗ dxj ⊗ dxk, Aijk =

F

2

[

12F2

]

yiyjyk

;

η := Aijkgjkdxi,

(

gjk)

=(

gij)−1

,

(2.3)

which are called the Cartan tensor and the Cartan form, respectively [9]. Each fibre of π∗T ∗Mhas a positively oriented orthonormal coframe {ωi} with ωn = ω and g =

∑

i (ωi)2 ∈

Γ(�2π∗T ∗M). Expand {ωi} as vijdx

j , whereby the stipulated orientation implies that v =

det(vij) =

√

det(gij).Define

δ

δxi:=

∂

∂xi−N

j

i

∂

∂yj, δyi =

1F

(

dyi +Nijdx

j)

, ωn+i = vijδy

j , (2.4)

where Nij := γijky

k −Aijkγ

kps(y

pys/F) and γijk are the formal Christoffel symbols of the secondkind for gij . Note that ω2n = d logF is dual with the radial vector yi(∂/∂yi), so it vanishes onthe projective tangent bundle SM. So {ωi,ωn+i} forms an orthonormal basis for T ∗(TM\{0})with respect to the Sasaki metric

g = gij(

x, y)

dxi ⊗ dxj + gij(

x, y)δyi

F⊗ δyj

F. (2.5)

It is well known that there exists the unique Chern connection c∇ on π∗TM withc∇(∂/∂xj) = ωi

j(∂/∂xi) and ωi

j = Γijkdxk, which satisfies the following structure equation:

dωi = ωj ∧ωij ,

ωij +ω

j

i = −2Aijaωa,

(2.6)

where ωa = ωan, Aija = A(ei, ej , ea). The Chern connection is torsion-free and almost

compatible with metric.

4 ISRN Geometry

The Berwald connection b∇ is also an important connection on π∗TM, which istorsion-free and given by

b∇ = c∇ + A equivalently bωij = ωi

j + Aijkω

k, (2.7)

where “·” denotes the covariant derivative along the Hilbert form. The one-form of theBerwald connection bωi

j satisfies

dωi = ωj ∧ bωij ,

bωij +

bωj

i = −2Aijaωa + 2Aijkω

k.(2.8)

The curvature 2-form of the Chern connection c∇ is given by

dωij −ωk

j ∧ωik = Ωi

j =12Ri

jklωk ∧ωl + Pi

jkaωk ∧ωa, (2.9)

where Rijkl

= −Rijlk, P i

jka= Pi

kja. For the Landsberg curvature Pi

jk:= Pi

njk, we have

Pijk = δilPljk = −Aijk, Pnjk = 0. (2.10)

Similarly, the curvature 2-form of the Berwald connection b∇ can also be expressed as

d bωij − bωk

j ∧ bωik =b Ωi

j =12

bRijklω

k ∧ωl + bP ijkaω

k ∧ωa, (2.11)

where bRijkl = − bRi

jlk,bP i

jka = bP ikja.

Next, we will give several definitions which will be used in the following.

Definition 2.1. For any X = Xi(∂/∂xi) ∈ π∗TM, the Ricci curvature under the Berwaldconnection in the direction X is given as

bRicM(

x, y,X)

:=1

〈X,X〉g∑

j,l,s

δksbRs

jlj

(

x, y)

XkXl. (2.12)

Obviously, if X = e, then the Ricci curvature is just the common scalar Ricci curvature.

Definition 2.2. For any X,Y ∈ π∗TM, the directional section curvature of M under the Chernconnection is given as

K(

x, y,X ∧ Y)

=Rijkl

(

x, y)

XiY jXkY l

〈X,X〉〈Y, Y〉 − 〈X,Y〉2. (2.13)

In general,K(x, y,X∧Y )/=K(x, y, Y ∧X). Particularly, ifM is the Riemannian manifold, thenK is the Riemannian section curvature.

ISRN Geometry 5

3. The Map between Finsler Manifolds

Let (M,F) and (˜M, ˜F) be Finsler manifolds of dimension n and m, respectively, and letφ : (M,F) → (˜M, ˜F) be a smooth map. F and ˜F induce the metrics g =

∑

i (ωi)2 and

g =∑

α (ωα)2, where {ωi} and {ωα} are the orthonormal one-form on TM and T ˜M, res-

pectively.In [5], Shen and Zhang give the tension field of the harmonic maps between real

Finsler manifolds. Recently, Shen and He [1] have simplified the tension field into thefollowing form:

d

dtE(

φ)

t|t=0 = − n

2cn−1

∫

SM

〈τ , V 〉gdVSM, (3.1)

where

τ :=(

˜∇ldφ)

(l) =1F2

τα∂

∂xα,

τα = φαijy

iyj − φαkG

k + ˜Gα,

(3.2)

where l = li(∂/∂xi), li = yi/F is the dual field of the Hilbert form, and Gk and ˜Gα are thegeodesic coefficients of (M,F) and (˜M, ˜F), respectively. Here

φαij =

∂2φα

∂xi∂xj, φα

i =∂φα

∂xi. (3.3)

From the formula (3.1), we have

Lemma 3.1 (see [5]). Let φ be harmonic map if and only if for any vector field V ∈ C(φ−1T ˜M),

∫

SM

〈τ , V 〉gdVSM = 0. (3.4)

φ is the strongly harmonic map if and only if τα = 0.

Let Φ : SM → S˜M be the map between the projective sphere bundles of M and˜M, which is induced by φ. It is easy to find that Φ∗ ◦ d = d ◦ Φ∗, φ∗(ωα) = φα

j ωj is just the

same asΦ∗(ωα) = φαj ω

j . Let {ωi,ωa+n} be the orthonormal frames of the dual bundle for SM,

and let {ωα, ωa+m} be the orthonormal frames of the dual bundle for S˜M. Then we have thefollowing.

6 ISRN Geometry

Proposition 3.2. Let Φ : SM → S˜M be the map between the projective sphere bundle of M and˜M. Then

Φ∗(ωα) = φαj ω

j ,

Φ∗(ωα+m) =12ταj ω

j + φαj ω

j+n.(3.5)

Obviously, if φ is a strongly harmonic map, then Φ∗(ωα+m) = φαj ω

j+n.

Proof. We will use natural frame to proof the Theorem. The relation between natural frameand moving frame satisfies

ωi = Cijdx

j , ωi+n = Dijδy

j , ωα = Aαβdu

β, ωα+m = Bαβδv

β, (3.6)

whereAαβ, Bα

β, Ci

j , Dij are orthonormal matrixes, and {dxi, δyi} and {dui, δvi} are the natural

bases of the dual bundle for SM and S˜M, respectively. Then we have

Φ∗(ωα) = Φ∗(

Aαβdu

β)

= AαβΦ

∗(

duβ)

= Aαβd(u

α ◦Φ)

= Aαβ

∂φα

∂xidxi = Aα

β

∂φα

∂xi

(

C−1)i

jωj := φα

i ωi,

Φ∗(ωα+m) = Φ∗(

Bαβδv

β)

= Bαβd(v

α ◦Φ) + BαβÑ

βγ

∂φγ

∂xjdxj

= Bαβ

(

∂2φα

∂xi∂xjyidxj +

∂φα

∂xjdyj

)

+ BαβÑ

βγ

∂φγ

∂xjdxj

= Bαβ

(

∂2φα

∂xi∂xjyi + Ñ

βγ

∂φγ

∂xj

)

(

C−1)j

kωk

+ Bαβ

∂φβ

∂xj

(

dyj +Nj

kdxk

)

− Bαβ

∂φβ

∂xjN

j

kdxk

= Bαβ

(

∂2φα

∂xi∂xjyi + Ñ

βγ

∂φγ

∂xj− ∂φβ

∂xlNl

j

)

(

C−1)j

kωk + Bα

β

∂φβ

∂xl

(

D−1)l

jωj+n

:=12ταj ω

j + φαj ω

j+n.

(3.7)

So, we have completed the proof of the proposition.

4. The Rigidity Theorem

In the following, let (M,F) be a Finsler manifold of dimension n, and let (˜M, ˜F) be a Berwaldmanifold of dimension m. Let φ : (M,F) → (˜M, ˜F) be a map with zero tension field, that

ISRN Geometry 7

is, strongly harmonic map. Because the Berwald connection on the Berwald manifold is thesame as the Chern connection, so we will use the Berwald connection on (M,F) and (˜M, ˜F).

Let

e(

φ)

=12

∑

α,i

(

φαi

)2 =12∥

∥φ∗∥

∥

2. (4.1)

Define

bφαi|jω

j + bφαi;aω

a := dφαi − φα

jbω

j

i + φβ

i Φ∗(

ωαβ

)

, (4.2)

bφαi|j|kω

k + bφαi|j;aω

a := d bφαi|j − bφα

l|jbωl

i − bφαi|l

bωlj +

bφβ

i|jΦ∗(

ωαβ

)

. (4.3)

Differentiating Φ∗ωα =∑

φαi ω

i and by Φ∗ ◦ d = d ◦Φ∗, we have

Φ∗dωα = dφαi ∧ωi + φα

i dωi. (4.4)

Substituting (2.8) and (4.2) into (4.4) yields

Φ∗(dωα) =(

bφαi|jω

j + bφαi;aω

a + φαjbω

j

i − φβ

i Φ∗(

ωαβ

))

∧ωi − φαibωi

k ∧ωk. (4.5)

On the other hand, since

Φ∗(dωα) = −Φ∗(

ωαβ ∧ ωβ

)

= −Φ∗(

ωαβ

)

∧Φ∗(

ωβ)

= −Φ∗(

ωαβ

)

∧ φβ

i ωi, (4.6)

from (4.5), we have

bφαi|jω

j ∧ωi + bφαi;aω

a ∧ωi = 0. (4.7)

That is,

bφαi;a = 0, bφα

i|j =bφα

j|i. (4.8)

Differentiating (4.2) and from (4.8), we get

bφαi|jdω

j +(

dbφαi|j)

∧ωj = −dφαj ∧ bω

j

i − φαj d

bωj

i + dφβ

i ∧Φ∗(

ωαβ

)

+ φβ

i Φ∗(

dωαβ

)

. (4.9)

8 ISRN Geometry

then by (2.8), (2.11), and (4.3), we have

(

bφαi|j|kω

k + bφαi|j;aω

a + bφαl|j

bωli +

bφαi|l

bωlj − bφ

β

i|jΦ∗(

ωαβ

)

)

∧ωj

= bφαi|j

bωj

k ∧ωk +(

− bφαj|lω

l − φαlbωl

j + φβ

jΦ∗(

ωαβ

))

∧ bωj

i

+ φαj

(

bωj

k ∧ bωki − bΩj

i

)

+ φβ

i Φ∗(

−ωαγ ∧ ω

γ

β + ˜Ωαβ

)

+(

bφβ

i|jωj + φ

β

lbωl

i − φγ

i Φ∗(

ωβγ

))

∧Φ∗(

ωαβ

)

.

(4.10)

Simplifying (4.10) yields

bφαi|j|kω

k ∧ωj + bφαi|j;aω

a ∧ωj = −φαjbΩj

i + φβ

i Φ∗˜Ωαβ. (4.11)

Note that

Φ∗(

˜Ωαβ

)

= Φ∗(

12˜Rαβγσω

γ ∧ ωσ + ˜Pαβγaω

γ ∧ ωa

)

=12˜Rαβγσφ

γ

kφσkω

k ∧ωl + ˜Pαβγaφ

γ

kφabω

k ∧ωb.

(4.12)

Substituting (2.11) and (4.12) into (4.11), by comparing the two sides of (4.11), we can get

bφαi|j|k − bφα

i|k|j = φαlbRl

ijk − ˜Rαβγσφ

β

i φγ

j φσk ,

bφαi|j;a = φα

lbP l

ija − ˜Pαβγaφ

β

i φγ

j φaa.

(4.13)

By Definitions 2.1 and 2.2, we have the following Bochner-type formula.

Theorem 4.1. Let (M,F) be a Finsler manifold, and let (˜M, ˜F) be a Berwald manifold. If the tensionfield of φ : (M,F) → (˜M, ˜F) is zero, then

∑

j

Sj|j =∣

∣∇dφ∣

∣

2 +∑

i,j

⟨


j

−∑

i,j

(

∣

∣φ∗ei∣

∣

2∣∣φ∗ej

∣

∣

2 − ⟨


φ∗ei, φ∗ej⟩

)

K˜M(

φ∗ei ∧ φ∗ej)

,

(4.14)

where S =∑

α,i,j φαibφα

i|jωj := Sjω

j . In particular, if M and ˜M are Riemannian manifolds, then∑

j Sj|j is Δe(φ).

ISRN Geometry 9

Proof. Letting e(φ) = 1/2∑

α,i (φαi )

2, then we have

de(

φ)

=∑

α,i

φαi dφ

αi . (4.15)

Substituting (4.2) into (4.15) yields

de(

φ)

=∑

α,i,j

(

φαibφα

i|jωj + φα

i φαjbω

j

i − φαi φ

β

i Φ∗(

ωαβ

))

. (4.16)

By (2.8), we can get

∑

α,i,j

φαi φ

αjbω

j

i =12

∑

α,i,j

φαi φ

αj

(

bωj

i +bωi

j

)

= −∑

α,i,j

φαi φ

αj Aijaω

a +∑

α,i,j

φαi φ

αj Aijkω

k.

(4.17)

From (2.6) and (3.5), we can obtain

∑

α,β,i

φαi φ

β

i Φ∗(

ωαβ

)

=12

∑

α,β,i

φαi φ

β

i Φ∗(

ωαβ + ω

βα

)

= −∑

α,β,i

˜Aαβaφαi φ

β

i φaaω

a.

(4.18)

Substituting (4.17) and (4.18) into (4.16), we have

de(

φ)

=∑

α,i,j

φαibφα

i|jωj −

∑

α,i,j

φαi φ

αj Aijaω

a

+∑

α,i,j

φαi φ

αj Aijkω

k +∑

α,β,i

˜Aαβaφαi φ

β

i φaaω

a.(4.19)

Defining S =∑

α,i,j φαibφα

i|jωj := Sjω

j , then we have

∑

j

Sj|j =∑

i

(

b∇eiSjωj)

(ei)

=∑

i

(dSi)(ei) −∑

i,j

Sjbω

j

i (ei)

=∑

i

(dSi)(ei) −∑

i,l,j

φαlbφα

l|jbω

j

i (ei).

(4.20)

ISRN Geometry 11

Substituting (4.23) and (4.26) into (4.22) yields

∑

j

Sj|j =∑

α,i,l

(

b φαl|i)2

+∑

α,i,l

φαl φ

αpbR

p

ili −∑

α,i,l

˜Rαβγσφ

αl φ

β

i φγ

l φσi . (4.27)

Then we get Theorem 4.1.

From Definitions 2.1 and 2.2, and Theorem 4.1, we can get the following rigiditytheorem

Theorem 4.2. Let M be a compact Finsler manifold of dimension n, and let ˜M be a Berwald manifoldof dimension m. Suppose a, b are positive constants, for any X ∈ π∗TM, bRic(X) ≥ a, and K

˜M ≤ b,whereK˜M is the directional section curvature of ˜M. Suppose the tension field of φ : M → ˜M is zeroand φ is nondegenerate. If

e(

φ) ≤ n

2(n − 1)a

b, (4.28)

then φ is a constant map or totally geodesic map. In particular, if e(φ) ≤ a/2b, then φ must be aconstant map.

Proof. From Theorem 4.1, we have

∑

j

Sj|j =∣

∣∇dφ∣

∣

2 +∑

i,j

⟨


j

−∑

i,j

(

∣

∣φ∗ei∣

∣

2∣∣φ∗ej

∣

∣

2 − ⟨


φ∗ei, φ∗ej⟩

)

K˜M(

φ∗ei, φ∗ej)

(4.29)

Diagonalizing 〈φ∗ei, φ∗ej〉 at a point (x, y) ∈ SM, then we have

⟨

φ∗ei, φ∗ej⟩

= λiδij . (4.30)

Fixing a point (x, y), then eigenvalues {λi} can be sorted as the following sequence:

λ1(

x, y) ≥ λ2

(

x, y) ≥ · · · ≥ λn

(

x, y)

> 0. (4.31)

So, we have

∑

j

Sj|j ≥∣

∣∇dφ∣

∣

2 − b

(

4e2(

φ) −

n∑

k=1

λ2k

)

+ 2ae(

φ)

. (4.32)

Then by the Schwarz inequality, we can get

∑

j

Sj|j ≥∣

∣∇dφ∣

∣

2 + 2e(

φ)

(

a − 2(n − 1)n

be(

φ)

)

. (4.33)

12 ISRN Geometry

Integrating the two sides of (4.33), we have

∣

∣∇dφ∣

∣

2 = 0, (4.34)

2e(

φ)

(

a − 2(n − 1)n

be(

φ)

)

= 0. (4.35)

If e(φ)/= 0, then φ is a totally geodesic map and e(φ) ≡ (2(n − 1)/n)b. In particular, if e(φ) ≤a/2b, then e(φ) < (n/2(n − 1))(a/b). So by (4.35), we have

e(

φ) ≡ 0, (4.36)

that is, φ is a constant map.

Acknowledgments

This work is supported by the NSF of China under Grant no. 11001069 and 11026105,the authors are also supported by the Hangdian Foundation KYS075608077. they thankanonymous referees for their valuable suggestions and pertinent criticisms.

References

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[2] Q. He and Y. B. Shen, “Some properties of harmonic maps for Finsler manifolds,” Houston Journal ofMathematics, vol. 33, no. 3, pp. 683–699, 2007.

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