The Energy Density Gap of Harmonic Maps between Finsler Manifolds

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.


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 metricmeasure 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 2 ISRN Geometry tension field from the Finsler manifold to the Berwald manifold, which generalize the results in 2 .
Precisely, we prove the following Bochner-type formula.where S α,i,j φ α i b φ α i|j ω j : S j ω j .In particular, if M and M are Riemannian manifolds, then j S j|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 manifold of dimension m.Suppose a, b are positive constants, for any X ∈ π * TM, b Ric X ≥ a, and K M ≤ b, where K M is the directional section curvature of M. Suppose the tension field of φ : 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 Section 2, 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.

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 Here and from now on, F y i , F y i y j denote ∂F/∂y i , ∂ 2 F/∂y i ∂y j , and so forth, and we will use the following convention of index range unless otherwise stated:

2.2
The canonical projection π : TM → M gives to a covector bundle π * T * M which has a global section ω F y i dx i called the Hilbert form, whose dual vector field is l y i /F ∂/∂x i l i ∂/∂x i , viewed as a global section of the pull-back bundle π * TM.We have the following important quantities: which are called the Cartan tensor and the Cartan form, respectively 9 .Each fibre of π * T * M has a positively oriented orthonormal coframe {ω i } with ω n ω and where N i j : γ i jk y k − A i jk γ k ps y p y s /F and γ i jk are the formal Christoffel symbols of the second kind for g ij .Note that ω 2n d log F is dual with the radial vector y i ∂/∂y i , so it vanishes on the projective tangent bundle SM.So {ω i , ω n i } forms an orthonormal basis for T * TM \ {0} with respect to the Sasaki metric It is well known that there exists the unique Chern connection c ∇ on π * TM with c ∇ ∂/∂x j ω i j ∂/∂x i and ω i j Γ i jk dx k , which satisfies the following structure equation: where ω a ω a n , A ija A e i , e j , e a .The Chern connection is torsion-free and almost compatible with metric.

ISRN Geometry
The Berwald connection b ∇ is also an important connection on π * TM, which is torsion-free and given by where "•" denotes the covariant derivative along the Hilbert form.The one-form of the Berwald connection b ω i j satisfies

2.8
The curvature 2-form of the Chern connection c ∇ is given by where R i jkl −R i jlk , P i jka P i kja .For the Landsberg curvature P i jk : P i njk , we have Similarly, the curvature 2-form of the Berwald connection b ∇ can also be expressed as Next, we will give several definitions which will be used in the following.Definition 2.1.For any X X i ∂/∂x i ∈ π * TM, the Ricci curvature under the Berwald connection in the direction X is given as 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 Chern connection is given as

2.13
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.

The Map between Finsler Manifolds
Let M, F and M, F be Finsler manifolds of dimension n and m, respectively, and let φ : M, F → M, F be a smooth map.F and F induce the metrics g i ω i 2 and g α ω α 2 , where {ω i } and { ω α } are the orthonormal one-form on TM and T M, 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: where where l l i ∂/∂x i , l i y i /F is the dual field of the Hilbert form, and G k and G α are the geodesic coefficients of M, F and M, F , respectively.Here 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 φ is the strongly harmonic map if and only if τ α 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 * ω α φ α j ω j is just the same as Φ * ω α φ α j ω j .Let {ω i , ω a n } be the orthonormal frames of the dual bundle for SM, and let { ω α , ω a m } be the orthonormal frames of the dual bundle for S M. Then we have the following.Proposition 3.2.Let Φ : SM → S M be the map between the projective sphere bundle of M and M. Then Φ * ω α m 1 2 τ α j ω j φ α j ω j n .
Proof.We will use natural frame to proof the Theorem.The relation between natural frame and moving frame satisfies where A α β , B α β , C i j , D i j are orthonormal matrixes, and {dx i , δy i } and {du i , δv i } are the natural bases of the dual bundle for SM and S M, respectively.Then we have

3.7
So, we have completed the proof of the proposition.

The Rigidity Theorem
In the following, let M, F be a Finsler manifold of dimension n, and let M, F be a Berwald manifold of dimension m.Let φ : M, F → M, F 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 M, F .Let Substituting 2.8 and 4.2 into 4.4 yields On the other hand, since That is,

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

4.13
By Definitions 2.1 and 2.2, we have the following Bochner-type formula. 4.17 From 2.6 and 3.5 , we can obtain A αβa φ α i φ β i φ a a ω a .

4.19
Defining S α,i,j φ α i b φ α i|j ω j : S j ω j , then we have from 4.8 , we get b φ α i|j dω j d b φ α i|j ∧ ω j −dφ α j ∧ b ω j i

FromTheorem 4 . 2 .
φ is a strongly harmonic map, from 3.2 and Lemma 3.1, we have that Definitions 2.1 and 2.2, and Theorem 4.1, we can get the following rigidity theorem Let M be a compact Finsler manifold of dimension n, and let M be a Berwald manifold of dimension m.Suppose a, b are positive constants, for any X ∈ π * TM, b Ric X ≥ a, and K M ≤ b, where K M is the directional section curvature of M. Suppose the tension field of φ : M → M e i , φ * e j φ * e i , φ * e j K M φ * e i ∧ φ * e j , Theorem 4.1.Let M, F be a Finsler manifold, and let M, F be a Berwald manifold.If the tension field of φ : M, F → M, F is zero, then * j : S j ω j .In particular, if M and M are Riemannian manifolds, then j S j|j is Δe φ .
is zero and φ is nondegenerate.If e j 2 − φ * e i , φ * e j φ * e i , φ * e j K M φ * e i , φ * e j 4.29 Diagonalizing φ * e i , φ * e j at a point x, y ∈ SM, then we have φ * e i , φ * e j λ i δ ij .4.30Fixing a point x, y , then eigenvalues {λ i } can be sorted as the following sequence:λ 1 x, y ≥ λ 2 x, y ≥ • • • ≥ λ n x, y > 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 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 Theorem 4.1, we have j S j|j ∇dφ * j S j|j ≥ ∇dφ