# The Energy Density Gap of Harmonic Maps between Finsler Manifolds

Authors: Jingwei Han; Yao-yong Yu; Jing Yu
Journal: ISRN Geometry
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Finsler manifolds are differential manifolds with Finsler metrics. Finsler metrics are Riemannian metrics but without quadratic restriction, which were firstly introduced by B. Riemann in 1854. Harmonic maps are important and interesting in both differential geometry and mathematical physics. Riemannian manifolds and Finsler manifolds are all metric-measure spaces, so we can study the harmonic map between Finsler manifolds by the theory of harmonic maps on general metric-measure spaces.By using the volume measure induced from the projective sphere bundle, harmonic maps 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 map from a real Finsler manifold to a Riemannian manifold. In [5], Shen and Zhang give the tension 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 mapping or totally geodesic mapping? This is an important and interesting issue in the study of harmonic maps, which is referred to as the rigidity theorem and studied by many people on the 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 tension field from the Finsler manifold to the Berwald manifold, which generalize the results in [2].Precisely, we prove the following Bochner-type formula.Theorem 1.1.
Let(M,F) be a Finsler manifold, and let (M̃,F̃) be a Berwald manifold. If the tension field of ϕ:(M,F)→(M̃,F̃) is zero, then
(1.1)∑jSj|j=|∇dϕ|2+∑i,j〈ϕ*ei,ϕ*ej〉bRji-∑i,j(|ϕ*ei|2|ϕ*ej|2-〈ϕ*ei,ϕ*ej〉〈ϕ*ei,ϕ*ej〉)KM̃(ϕ*ei∧ϕ*ej),
where 𝒮=∑α,i,jϕiαbϕi|jαωj:=𝒮jωj. In particular, if M and M̃ are Riemannian manifolds, then ∑j𝒮j|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 dimensionn, and let M̃ be a Berwald manifold of dimension m. Suppose a,b are positive constants, for any X∈π*TM, bRic(X)≥a, and KM̃≤b, where KM̃ is the directional section curvature of M̃. Suppose the tension field of ϕ:M→M̃ is zero and ϕ is nondegenerate. If
(1.2)e(ϕ)≤n2(n-1)ab,
then ϕ is a constant map or totally geodesic map. In particular, if e(ϕ)≤a/2b, then ϕ must be a constant map.Some technical terms above will be explained below. The contents of the paper are arranged as follows. In Section2, some fundamental definitions and formulas which are necessary for the present paper are given. In Section 3, we consider the map between Finsler manifolds and get a pull-back formula. In Section 4, a Bochner-type formula from the Finsler manifold to the Berwald manifold is shown. Finally, by using the Bochner type formula, we obtain a rigidity theorem.


## 2. Finsler Manifold

LetM 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 formg is positively definite, where(2.1)g=gij(x,y)dxi⊗dxj,gij=12(F2)yiyj.Here and from now on,[F]yi, [F]yiyj denote ∂F/∂yi, ∂2F/∂yi∂yj, and so forth, and we will use the following convention of index range unless otherwise stated: (2.2)1≤i,j,k,…≤n,1≤a,b,c,…≤n-1,a̅=n+a,n=dim⁡M;1≤α,β,γ⋯≤m,m=dim⁡M̃.The canonical projectionπ:TM→M gives to a covector bundle π*T*M which has 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. We have the following important quantities: (2.3)A:=Aijkdxi⊗dxj⊗dxk,Aijk=F2[12F2]yiyjyk;η:=Aijkgjkdxi,(gjk)=(gij)-1,
which are called the Cartan tensor and the Cartan form, respectively [9]. Each fibre of π*T*M has a positively oriented orthonormal coframe {ωi} with ωn=ω and g=∑i(ωi)2∈Γ(⊙2π*T*M). Expand {ωi} as vjidxj, whereby the stipulated orientation implies that v=det(vji)=det(gij).Define(2.4)δδxi:=∂∂xi-Nij∂∂yj,δyi=1F(dyi+Njidxj),ωn+i=vjiδyj,
where Nji:=γjkiyk-Ajkiγpsk(ypys/F) and γjki are the formal Christoffel symbols of the second kind for gij. Note that ω2n=dlogF is dual with the radial vector yi(∂/∂yi), so it vanishes on the projective tangent bundle SM. So {ωi,ωn+i} forms an orthonormal basis for T*(TM∖{0}) with respect to the Sasaki metric (2.5)ĝ=gij(x,y)dxi⊗dxj+gij(x,y)δyiF⊗δyjF.It is well known that there exists the unique Chern connectionc∇ on π*TM with c∇(∂/∂xj)=ωji(∂/∂xi) and ωji=Γjkidxk, which satisfies the following structure equation: (2.6)dωi=ωj∧ωji,ωji+ωij=-2Aijaωa̅,
where ωa̅=ωna, Aija=A(ei,ej,ea). The Chern connection is torsion-free and almost compatible with metric.The Berwald connectionb∇ is also an important connection on π*TM, which is torsion-free and given by(2.7)b∇=c∇+Ȧequivalentlybωji=ωji+Ȧjkiωk,
where “·” denotes the covariant derivative along the Hilbert form. The one-form of the Berwald connection bωji satisfies (2.8)dωi=ωj∧bωji,bωji+bωij=-2Aijaωa̅+2Ȧijkωk.The curvature 2-form of the Chern connectionc∇ is given by (2.9)dωji-ωjk∧ωki=Ωji=12Rjkliωk∧ωl+Pjkaiωk∧ωa̅,
where Rjkli=-Rjlki,Pjkai=Pkjai. For the Landsberg curvature Pjki:=Pnjki, we have (2.10)Pijk=δilPjkl=-Ȧijk,Pnjk=0.Similarly, the curvature 2-form of the Berwald connectionb∇ can also be expressed as (2.11)dbωji-bωjk∧bωki=bΩji=12bRjkliωk∧ωl+bPjkaiωk∧ωa̅,
where bRjkli=-bRjlki,bPjkai=bPkjai.Next, we will give several definitions which will be used in the following.Definition 2.1.
For anyX=Xi(∂/∂xi)∈π*TM, the Ricci curvature under the Berwald connection in the direction X is given as
(2.12)bRicM(x,y,X):=1〈X,X〉g∑j,l,sδksbRjljs(x,y)XkXl.
Obviously, if X=ê, then the Ricci curvature is just the common scalar Ricci curvature.Definition 2.2.
For anyX,Y∈π*TM, the directional section curvature of M under the Chern connection is given as
(2.13)K(x,y,X∧Y)=Rijkl(x,y)XiYjXkYl〈X,X〉〈Y,Y〉-〈X,Y〉2.
In general, K(x,y,X∧Y)≠K(x,y,Y∧X). Particularly, if M is the Riemannian manifold, then K is the Riemannian section curvature.


## 3. The Map between Finsler Manifolds

Let(M,F) and (M̃,F̃) be Finsler manifolds of dimension n and m, respectively, and let ϕ:(M,F)→(M̃,F̃) be a smooth map. F and F̃ induce the metrics g=∑i(ωi)2 and g̃=∑α(ω̃α)2, where {ωi} and {ω̃α} are the orthonormal one-form on TM and TM̃, respectively.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 the following form: (3.1)ddtE(ϕ)t∣t=0=-n2cn-1∫SM〈τ̃,V〉g̃dVSM,
where (3.2)τ̃:=(∇̃ldϕ)(l)=1F2τ̃α∂∂x̃α,τ̃α=ϕijαyiyj-ϕkαGk+G̃α,
where l=li(∂/∂xi),li=yi/F is the dual field of the Hilbert form, and Gk and G̃α are the geodesic coefficients of (M,F) and (M̃,F̃), respectively. Here (3.3)ϕijα=∂2ϕα∂xi∂xj,ϕiα=∂ϕα∂xi.
From the formula (3.1), we haveLemma 3.1 (see [5]).
Letϕ be harmonic map if and only if for any vector field V∈𝒞(ϕ-1TM̃),
(3.4)∫SM〈τ̃,V〉g̃dVSM=0.
ϕ is the strongly harmonic map if and only if τ̃α=0.LetΦ:SM→SM̃ be the map between the projective sphere bundles of M and M̃, which is induced by ϕ. It is easy to find that Φ*∘d=d∘Φ*, ϕ*(ω̃α)=ϕjαωj is just the same as Φ*(ω̃α)=ϕjαωj. Let {ωi,ωa+n} be the orthonormal frames of the dual bundle for SM, and let {ω̃α,ω̃𝔞+m} be the orthonormal frames of the dual bundle for SM̃. Then we have the following.Proposition 3.2.
LetΦ:SM→SM̃ be the map between the projective sphere bundle of M and M̃. Then
(3.5)Φ*(ω̃α)=ϕjαωj,Φ*(ω̃α+m)=12τ̃jαωj+ϕjαωj+n.
Obviously, if ϕ is a strongly harmonic map, then Φ*(ωα+m)=ϕjαωj+n.Proof.
We will use natural frame to proof the Theorem. The relation between natural frame and moving frame satisfies(3.6)ωi=Cjidxj,ωi+n=Djiδyj,ω̃α=Aβαduβ,ω̃α+m=Bβαδvβ,
where Aβα,Bβα,Cji,Dji are orthonormal matrixes, and {dxi,δyi} and {dui,δvi} are the natural bases of the dual bundle for SM and SM̃, respectively. Then we have
(3.7)Φ*(ω̃α)=Φ*(Aβαduβ)=AβαΦ*(duβ)=Aβαd(uα∘Φ)=Aβα∂ϕα∂xidxi=Aβα∂ϕα∂xi(C-1)jiωj:=ϕiαωi,Φ*(ω̃α+m)=Φ*(Bβαδvβ)=Bβαd(vα∘Φ)+BβαÑγβ∂ϕγ∂xjdxj=Bβα(∂2ϕα∂xi∂xjyidxj+∂ϕα∂xjdyj)+BβαÑγβ∂ϕγ∂xjdxj=Bβα(∂2ϕα∂xi∂xjyi+Ñγβ∂ϕγ∂xj)(C-1)kjωk+Bβα∂ϕβ∂xj(dyj+Nkjdxk)-Bβα∂ϕβ∂xjNkjdxk=Bβα(∂2ϕα∂xi∂xjyi+Ñγβ∂ϕγ∂xj-∂ϕβ∂xlNjl)(C-1)kjωk+Bβα∂ϕβ∂xl(D-1)jlωj+n:=12τ̃jαωj+ϕjαωj+n.
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 (M̃,F̃) be a Berwald manifold of dimension m. Let ϕ:(M,F)→(M̃,F̃) be a map with zero tension field, that is, strongly harmonic map. Because the Berwald connection on the Berwald manifold is the same as the Chern connection, so we will use the Berwald connection on (M,F) and (M̃,F̃).Let(4.1)e(ϕ)=12∑α,i(ϕiα)2=12‖ϕ*‖2.
Define (4.2)bϕi|jαωj+bϕi;aαωa̅:=dϕiα-ϕjαbωij+ϕiβΦ*(ω̃βα),(4.3)bϕi|j|kαωk+bϕi|j;aαωa̅:=dbϕi|jα-bϕl|jαbωil-bϕi|lαbωjl+bϕi|jβΦ*(ω̃βα).
Differentiating  Φ*ω̃α=∑ϕiαωi and by Φ*∘d=d∘Φ*, we have (4.4)Φ*dω̃α=dϕiα∧ωi+ϕiαdωi.
Substituting (2.8) and (4.2) into (4.4) yields (4.5)Φ*(dω̃α)=(bϕi|jαωj+bϕi;aαωa̅+ϕjαbωij-ϕiβΦ*(ω̃βα))∧ωi-ϕiαbωki∧ωk.
On the other hand, since (4.6)Φ*(dω̃α)=-Φ*(ω̃βα∧ω̃β)=-Φ*(ω̃βα)∧Φ*(ω̃β)=-Φ*(ω̃βα)∧ϕiβωi,
from (4.5), we have (4.7)bϕi|jαωj∧ωi+bϕi;aαωa̅∧ωi=0.
That is, (4.8)bϕi;aα=0,bϕi|jα=bϕj|iα.
Differentiating (4.2) and from (4.8), we get (4.9)bϕi|jαdωj+(dbϕi|jα)∧ωj=-dϕjα∧bωij-ϕjαdbωij+dϕiβ∧Φ*(ω̃βα)+ϕiβΦ*(dω̃βα).
then by (2.8), (2.11), and (4.3), we have (4.10)(bϕi|j|kαωk+bϕi|j;aαωa̅+bϕl|jαbωil+bϕi|lαbωjl-bϕi|jβΦ*(ω̃βα))∧ωj=bϕi|jαbωkj∧ωk+(-bϕj|lαωl-ϕlαbωjl+ϕjβΦ*(ω̃βα))∧bωij+ϕjα(bωkj∧bωik-bΩij)+ϕiβΦ*(-ω̃γα∧ω̃βγ+Ω̃βα)+(bϕi|jβωj+ϕlβbωil-ϕiγΦ*(ω̃γβ))∧Φ*(ω̃βα).
Simplifying (4.10) yields (4.11)bϕi|j|kαωk∧ωj+bϕi|j;aαωa̅∧ωj=-ϕjαbΩij+ϕiβΦ*Ω̃βα.
Note that (4.12)Φ*(Ω̃βα)=Φ*(12R̃βγσαω̃γ∧ω̃σ+P̃βγaαω̃γ∧ω̃a̅)=12R̃βγσαϕkγϕkσωk∧ωl+P̃βγaαϕkγϕbaωk∧ωb̅.
Substituting (2.11) and (4.12) into (4.11), by comparing the two sides of (4.11), we can get (4.13)bϕi|j|kα-bϕi|k|jα=ϕlαbRijkl-R̃βγσαϕiβϕjγϕkσ,bϕi|j;aα=ϕlαbPijal-P̃βγaαϕiβϕjγϕaa.By Definitions2.1 and 2.2, we have the following Bochner-type formula.Theorem 4.1.
Let(M,F) be a Finsler manifold, and let (M̃,F̃) be a Berwald manifold. If the tension field of ϕ:(M,F)→(M̃,F̃) is zero, then
(4.14)∑jSj∣j=|∇dϕ|2+∑i,j〈ϕ*ei,ϕ*ej〉bRji-∑i,j(|ϕ*ei|2|ϕ*ej|2-〈ϕ*ei,ϕ*ej〉〈ϕ*ei,ϕ*ej〉)KM̃(ϕ*ei∧ϕ*ej),
where 𝒮=∑α,i,jϕiαbϕi|jαωj:=𝒮jωj. In particular, if M and M̃ are Riemannian manifolds, then ∑j𝒮j|j is Δe(ϕ).Proof.
Lettinge(ϕ)=1/2∑α,i(ϕiα)2, then we have
(4.15)de(ϕ)=∑α,iϕiαdϕiα.
Substituting (4.2) into (4.15) yields
(4.16)de(ϕ)=∑α,i,j(ϕiαbϕi|jαωj+ϕiαϕjαbωij-ϕiαϕiβΦ*(ω̃βα)).
By (2.8), we can get
(4.17)∑α,i,jϕiαϕjαbωij=12∑α,i,jϕiαϕjα(bωij+bωji)=-∑α,i,jϕiαϕjαAijaωa̅+∑α,i,jϕiαϕjαȦijkωk.
From (2.6) and (3.5), we can obtain
(4.18)∑α,β,iϕiαϕiβΦ*(ω̃βα)=12∑α,β,iϕiαϕiβΦ*(ω̃βα+ω̃αβ)=-∑α,β,iÃαβaϕiαϕiβϕaaωa̅.
Substituting (4.17) and (4.18) into (4.16), we have
(4.19)de(ϕ)=∑α,i,jϕiαbϕi|jαωj-∑α,i,jϕiαϕjαAijaωa̅+∑α,i,jϕiαϕjαȦijkωk+∑α,β,iÃαβaϕiαϕiβϕaaωa̅.
Defining 𝒮=∑α,i,jϕiαbϕi|jαωj:=𝒮jωj, then we have
(4.20)∑jSj∣j=∑i(b∇êiSjωj)(êi)=∑i(dSi)(êi)-∑i,jSjbωij(êi)=∑i(dSi)(êi)-∑i,l,jϕlαbϕl|jαbωij(êi).
On the other hand, from (4.2) and (4.3), we have
(4.21)∑i(dSi)(êi)=∑α,i,ld(ϕlαbϕl∣iα)(êi)=∑α,i,lbϕl|iα(dϕlα)(êi)+∑α,i,lϕlαdbϕl|iα(êi)=∑α,i,lbϕl|iα(bϕl|iα+ϕpαbωlp(êi)-ϕlβΦ*(ω̃βα)(êi))+∑α,i,lϕlα(bϕl|i|iα+bϕl|pαbωip(êi)+bϕp|iαbωlp(êi)-bϕl|iβΦ*(ω̃βα)(êi))=∑α,i,l(bϕl|iα)2+∑α,i,lϕlαbϕl|i|iα+∑α,i,lbϕl|iαϕpαbωlp(êi)+∑α,i,lϕlαbϕp|iαbωlp(êi)+∑α,i,lϕlαbϕl|pαbωip(êi).
Substituting (4.21) into (4.20) and by (2.8), we can get
(4.22)∑jSi|i=∑α,i,l(bϕl|iα)2+∑α,i,lϕlαbϕl|i|iα+∑α,i,lϕlαbϕp|iα(bωlp+bωpl)(êi)=∑α,i,l(bϕl|iα)2+∑α,i,lϕlαbϕl|i|iα+2∑α,i,lϕlαbϕp|iαȦpli.
From (4.3), we have
(4.23)∑α,i,lϕlαbϕl|i|iα=∑α,i,lϕlαbϕi|i|lα+∑α,i,lϕlαϕpαbRilip-∑α,i,lR̃βγσαϕlαϕiβϕlγϕiσ.
Because ϕ is a strongly harmonic map, from (3.2) and Lemma 3.1, we have that
(4.24)∑ibϕi|iα=0,
Then
(4.25)0=∑idbϕi|iα(êl)=∑α,i,lbϕi|i|lα+∑α,i,lbϕp|iαbωip(êl)+∑α,i,lbϕi|pαbωip(êl).
So
(4.26)∑α,i,lϕlαbϕi|i|lα=-∑α,i,lϕlαbϕp|iα(bωip+bωpi)(êl)=-2∑α,i,lϕlαbϕp|iαȦpil.
Substituting (4.23) and (4.26) into (4.22) yields
(4.27)∑jSj|j=∑α,i,l(bϕl|iα)2+∑α,i,lϕlαϕpαbRilip-∑α,i,lR̃βγσαϕlαϕiβϕlγϕiσ.
Then we get Theorem 4.1.From Definitions2.1 and 2.2, and Theorem 4.1, we can get the following rigidity theoremTheorem 4.2.
Let M be a compact Finsler manifold of dimensionn, and let M̃ be a Berwald manifold of dimension m. Suppose a,b are positive constants, for any X∈π*TM, bRic(X)≥a, and KM̃≤b, where KM̃ is the directional section curvature of M̃. Suppose the tension field of ϕ:M→M̃ is zero and ϕ is nondegenerate. If
(4.28)e(ϕ)≤n2(n-1)ab,
then ϕ is a constant map or totally geodesic map. In particular, if e(ϕ)≤a/2b, then ϕ must be a constant map.Proof.
From Theorem4.1, we have
(4.29)∑jSj|j=|∇dϕ|2+∑i,j〈ϕ*ei,ϕ*ej〉bRji-∑i,j(|ϕ*ei|2|ϕ*ej|2-〈ϕ*ei,ϕ*ej〉〈ϕ*ei,ϕ*ej〉)KM̃(ϕ*ei,ϕ*ej)
Diagonalizing   〈ϕ*ei,ϕ*ej〉 at a point (x,y)∈SM, then we have
(4.30)〈ϕ*ei,ϕ*ej〉=λiδij.
Fixing a point (x,y), then eigenvalues {λi} can be sorted as the following sequence:
(4.31)λ1(x,y)≥λ2(x,y)≥⋯≥λn(x,y)>0.
So, we have
(4.32)∑jSj|j≥|∇dϕ|2-b(4e2(ϕ)-∑k=1nλk2)+2ae(ϕ).
Then by the Schwarz inequality, we can get
(4.33)∑jSj|j≥|∇dϕ|2+2e(ϕ)(a-2(n-1)nbe(ϕ)).
Integrating the two sides of (4.33), we have
(4.34)|∇dϕ|2=0,(4.35)2e(ϕ)(a-2(n-1)nbe(ϕ))=0.
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
(4.36)e(ϕ)≡0,
that is, ϕ is a constant map.



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