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.
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 . In  Mo considered the energy functional and the Euler-Lagrange operator of a smooth map from a real Finsler manifold to a Riemannian manifold. In , Shen and Zhang give the tension field of the harmonic maps between Finsler manifolds. Recently, Shen and He  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 .
Precisely, we prove the following Bochner-type formula.
Theorem 1.1. Let be a Finsler manifold, and let be a Berwald manifold. If the tension field of is zero, then where . In particular, if and are Riemannian manifolds, then is .
Moreover, by using the formula we also prove the following rigidity theorem.
Theorem 1.2. Let M be a compact Finsler manifold of dimension , and let be a Berwald manifold of dimension . Suppose are positive constants, for any , , and , where is the directional section curvature of . Suppose the tension field of is zero and is nondegenerate. If then is a constant map or totally geodesic map. In particular, if , then 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.
2. Finsler Manifold
Let be an -dimension smooth manifold, and let be the natural projection. A Finsler metric on is a function satisfying the following properties:(i) is smooth on ;(ii) for all ;(iii)the induced quadratic form is positively definite, where
Here and from now on, , denote , , and so forth, and we will use the following convention of index range unless otherwise stated:
The canonical projection gives to a covector bundle which has a global section called the Hilbert form, whose dual vector field is , viewed as a global section of the pull-back bundle . We have the following important quantities: which are called the Cartan tensor and the Cartan form, respectively . Each fibre of has a positively oriented orthonormal coframe with and . Expand as , whereby the stipulated orientation implies that .
Define where and are the formal Christoffel symbols of the second kind for . Note that is dual with the radial vector , so it vanishes on the projective tangent bundle . So forms an orthonormal basis for with respect to the Sasaki metric
It is well known that there exists the unique Chern connection on with and , which satisfies the following structure equation: where , . The Chern connection is torsion-free and almost compatible with metric.
The Berwald connection is also an important connection on , which is torsion-free and given by where “·” denotes the covariant derivative along the Hilbert form. The one-form of the Berwald connection satisfies
The curvature 2-form of the Chern connection is given by where . For the Landsberg curvature , we have
Similarly, the curvature 2-form of the Berwald connection can also be expressed as where .
Next, we will give several definitions which will be used in the following.
Definition 2.1. For any , the Ricci curvature under the Berwald connection in the direction is given as Obviously, if , then the Ricci curvature is just the common scalar Ricci curvature.
Definition 2.2. For any , the directional section curvature of under the Chern connection is given as In general, . Particularly, if is the Riemannian manifold, then is the Riemannian section curvature.
3. The Map between Finsler Manifolds
Let and be Finsler manifolds of dimension and , respectively, and let be a smooth map. and induce the metrics and , where and are the orthonormal one-form on and , respectively.
In , Shen and Zhang give the tension field of the harmonic maps between real Finsler manifolds. Recently, Shen and He  have simplified the tension field into the following form: where where is the dual field of the Hilbert form, and and are the geodesic coefficients of and , respectively. Here From the formula (3.1), we have
Lemma 3.1 (see ). Let be harmonic map if and only if for any vector field , is the strongly harmonic map if and only if .
Let be the map between the projective sphere bundles of and , which is induced by . It is easy to find that , is just the same as . Let be the orthonormal frames of the dual bundle for , and let be the orthonormal frames of the dual bundle for . Then we have the following.
Proposition 3.2. Let be the map between the projective sphere bundle of and . Then Obviously, if is a strongly harmonic map, then .
Proof. We will use natural frame to proof the Theorem. The relation between natural frame and moving frame satisfies where are orthonormal matrixes, and and are the natural bases of the dual bundle for and , respectively. Then we have So, we have completed the proof of the proposition.
4. The Rigidity Theorem
In the following, let be a Finsler manifold of dimension , and let be a Berwald manifold of dimension . Let 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 and .
Let Define Differentiating and by , we have Substituting (2.8) and (4.2) into (4.4) yields On the other hand, since from (4.5), we have That is, Differentiating (4.2) and from (4.8), we get then by (2.8), (2.11), and (4.3), we have Simplifying (4.10) yields Note that Substituting (2.11) and (4.12) into (4.11), by comparing the two sides of (4.11), we can get
Theorem 4.1. Let be a Finsler manifold, and let be a Berwald manifold. If the tension field of is zero, then where . In particular, if and are Riemannian manifolds, then is .
Proof. Letting , then we have Substituting (4.2) into (4.15) yields By (2.8), we can get From (2.6) and (3.5), we can obtain Substituting (4.17) and (4.18) into (4.16), we have Defining , then we have On the other hand, from (4.2) and (4.3), we have Substituting (4.21) into (4.20) and by (2.8), we can get From (4.3), we have Because is a strongly harmonic map, from (3.2) and Lemma 3.1, we have that Then So Substituting (4.23) and (4.26) into (4.22) yields Then we get Theorem 4.1.
Theorem 4.2. Let M be a compact Finsler manifold of dimension , and let be a Berwald manifold of dimension . Suppose are positive constants, for any , , and , where is the directional section curvature of . Suppose the tension field of is zero and is nondegenerate. If then is a constant map or totally geodesic map. In particular, if , then must be a constant map.
Proof. From Theorem 4.1, we have Diagonalizing at a point , then we have Fixing a point , then eigenvalues can be sorted as the following sequence: So, we have Then by the Schwarz inequality, we can get Integrating the two sides of (4.33), we have If , then is a totally geodesic map and . In particular, if , then . So by (4.35), we have that is, is a constant map.
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 thank anonymous referees for their valuable suggestions and pertinent criticisms.
D. Bao, S. S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, vol. 200 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.View at: Zentralblatt MATH