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.

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 [1ā€“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 (š‘€,š¹) be a Finsler manifold, and let (ī‚‹ī‚š‘€,š¹) be a Berwald manifold. If the tension field of ī‚‹ī‚šœ™āˆ¶(š‘€,š¹)ā†’(š‘€,š¹) is zero, then ī“š‘—š’®š‘—|š‘—=||||āˆ‡š‘‘šœ™2+ī“š‘–,š‘—ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬š‘š‘…š‘–š‘—āˆ’ī“š‘–,š‘—ī‚€||šœ™āˆ—š‘’š‘–||2||šœ™āˆ—š‘’š‘—||2āˆ’ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—šœ™ī¬ī«āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬ī‚š¾ī‚‹š‘€ī€·šœ™āˆ—š‘’š‘–āˆ§šœ™āˆ—š‘’š‘—ī€ø,(1.1) where āˆ‘š’®=š›¼,š‘–,š‘—šœ™š›¼š‘–š‘šœ™š›¼š‘–|š‘—šœ”š‘—āˆ¶=š’®š‘—šœ”š‘—. In particular, if š‘€ and ī‚‹š‘€ are Riemannian manifolds, then āˆ‘š‘—š’®š‘—|š‘— is Ī”š‘’(šœ™).

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

Theorem 1.2. Let M be a compact Finsler manifold of dimension š‘›, and let ī‚‹š‘€ be a Berwald manifold of dimension š‘š. Suppose š‘Ž,š‘ are positive constants, for any š‘‹āˆˆšœ‹āˆ—š‘‡š‘€, š‘Ric(š‘‹)ā‰„š‘Ž, and š¾ī‚‹š‘€ā‰¤š‘, where š¾ī‚‹š‘€ is the directional section curvature of ī‚‹š‘€. Suppose the tension field of ī‚‹š‘€šœ™āˆ¶š‘€ā†’ is zero and šœ™ is nondegenerate. If š‘’š‘›(šœ™)ā‰¤š‘Ž2(š‘›āˆ’1)š‘,(1.2) then šœ™ is a constant map or totally geodesic map. In particular, if š‘’(šœ™)ā‰¤š‘Ž/2š‘, then šœ™ 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 š‘€ be an š‘›-dimension smooth manifold, and let šœ‹āˆ¶š‘‡š‘€ā†’š‘€ be the natural projection. A Finsler metric on š‘€ is a function š¹āˆ¶š‘‡š‘€ā†’[0,āˆž) satisfying the following properties:(i)š¹ is smooth on š‘‡š‘€ā§µ{0};(ii)š¹(š‘„,šœ†š‘¦)=šœ†š¹(š‘„,š‘¦) for all šœ†>0;(iii)the induced quadratic form š‘” is positively definite, where š‘”=š‘”š‘–š‘—(š‘„,š‘¦)š‘‘š‘„š‘–āŠ—š‘‘š‘„š‘—,š‘”š‘–š‘—=12ī€·š¹2ī€øš‘¦š‘–š‘¦š‘—.(2.1)

Here and from now on, [š¹]š‘¦š‘–, [š¹]š‘¦š‘–š‘¦š‘— denote šœ•F/šœ•š‘¦š‘–, šœ•2š¹/šœ•š‘¦š‘–šœ•š‘¦š‘—, and so forth, and we will use the following convention of index range unless otherwise stated: 1ā‰¤š‘–,š‘—,š‘˜,ā€¦ā‰¤š‘›,1ā‰¤š‘Ž,š‘,š‘,ā€¦ā‰¤š‘›āˆ’1,ī‚‹š‘Ž=š‘›+š‘Ž,š‘›=dimš‘€;1ā‰¤š›¼,š›½,š›¾ā‹Æā‰¤š‘š,š‘š=dimš‘€.(2.2)

The canonical projection šœ‹āˆ¶š‘‡š‘€ā†’š‘€ gives to a covector bundle šœ‹āˆ—š‘‡āˆ—š‘€ which has a global section šœ”=[š¹]š‘¦š‘–š‘‘š‘„š‘– called the Hilbert form, whose dual vector field is š‘™=(š‘¦š‘–/š¹)(šœ•/šœ•š‘„š‘–)=š‘™š‘–(šœ•/šœ•š‘„š‘–), viewed as a global section of the pull-back bundle šœ‹āˆ—š‘‡š‘€. We have the following important quantities: š“āˆ¶=š“š‘–š‘—š‘˜š‘‘š‘„š‘–āŠ—š‘‘š‘„š‘—āŠ—š‘‘š‘„š‘˜,š“š‘–š‘—š‘˜=š¹2ī‚ƒ12š¹2ī‚„š‘¦š‘–š‘¦š‘—š‘¦š‘˜;šœ‚āˆ¶=š“š‘–š‘—š‘˜š‘”š‘—š‘˜š‘‘š‘„š‘–,ī€·š‘”š‘—š‘˜ī€ø=ī€·š‘”š‘–š‘—ī€øāˆ’1,(2.3) which are called the Cartan tensor and the Cartan form, respectively [9]. Each fibre of šœ‹āˆ—š‘‡āˆ—š‘€ has a positively oriented orthonormal coframe {šœ”š‘–} with šœ”š‘›=šœ” and āˆ‘š‘”=š‘–(šœ”š‘–)2āˆˆĪ“(āŠ™2šœ‹āˆ—š‘‡āˆ—š‘€). Expand {šœ”š‘–} as š‘£š‘–š‘—š‘‘š‘„š‘—, whereby the stipulated orientation implies that š‘£=det(š‘£š‘–š‘—āˆš)=det(š‘”š‘–š‘—).

Define š›æš›æš‘„š‘–šœ•āˆ¶=šœ•š‘„š‘–āˆ’š‘š‘—š‘–šœ•šœ•š‘¦š‘—,š›æš‘¦š‘–=1š¹ī‚€š‘‘š‘¦š‘–+š‘š‘–š‘—š‘‘š‘„š‘—ī‚,šœ”š‘›+š‘–=š‘£š‘–š‘—š›æš‘¦š‘—,(2.4) where š‘š‘–š‘—āˆ¶=š›¾š‘–š‘—š‘˜š‘¦š‘˜āˆ’š“š‘–š‘—š‘˜š›¾š‘˜š‘š‘ (š‘¦š‘š‘¦š‘ /š¹) and š›¾š‘–š‘—š‘˜ are the formal Christoffel symbols of the second kind for š‘”š‘–š‘—. Note that šœ”2š‘›=š‘‘logš¹ is dual with the radial vector š‘¦š‘–(šœ•/šœ•š‘¦š‘–), so it vanishes on the projective tangent bundle š‘†š‘€. So {šœ”š‘–,šœ”š‘›+š‘–} forms an orthonormal basis for š‘‡āˆ—(š‘‡š‘€ā§µ{0}) with respect to the Sasaki metric Ģ‚š‘”=š‘”š‘–š‘—(š‘„,š‘¦)š‘‘š‘„š‘–āŠ—š‘‘š‘„š‘—+š‘”š‘–š‘—(š‘„,š‘¦)š›æš‘¦š‘–š¹āŠ—š›æš‘¦š‘—š¹.(2.5)

It is well known that there exists the unique Chern connection š‘āˆ‡ on šœ‹āˆ—š‘‡š‘€ with š‘āˆ‡(šœ•/šœ•š‘„š‘—)=šœ”š‘–š‘—(šœ•/šœ•š‘„š‘–) and šœ”š‘–š‘—=Ī“š‘–š‘—š‘˜š‘‘š‘„š‘˜, which satisfies the following structure equation: š‘‘šœ”š‘–=šœ”š‘—āˆ§šœ”š‘–š‘—,šœ”š‘–š‘—+šœ”š‘—š‘–=āˆ’2š“š‘–š‘—š‘Žšœ”š‘Ž,(2.6) where šœ”š‘Ž=šœ”š‘Žš‘›, š“š‘–š‘—š‘Ž=š“(š‘’š‘–,š‘’š‘—,š‘’š‘Ž). The Chern connection is torsion-free and almost compatible with metric.

The Berwald connection š‘āˆ‡ is also an important connection on šœ‹āˆ—š‘‡š‘€, which is torsion-free and given byš‘āˆ‡=š‘Ģ‡āˆ‡+š“equivalentlyš‘šœ”š‘–š‘—=šœ”š‘–š‘—+Ģ‡š“š‘–š‘—š‘˜šœ”š‘˜,(2.7) where ā€œĀ·ā€ denotes the covariant derivative along the Hilbert form. The one-form of the Berwald connection š‘šœ”š‘–š‘— satisfies š‘‘šœ”š‘–=šœ”š‘—āˆ§š‘šœ”š‘–š‘—,š‘šœ”š‘–š‘—+š‘šœ”š‘—š‘–=āˆ’2š“š‘–š‘—š‘Žšœ”š‘ŽĢ‡š“+2š‘–š‘—š‘˜šœ”š‘˜.(2.8)

The curvature 2-form of the Chern connection š‘āˆ‡ is given by š‘‘šœ”š‘–š‘—āˆ’šœ”š‘˜š‘—āˆ§šœ”š‘–š‘˜=Ī©š‘–š‘—=12š‘…š‘–š‘—š‘˜š‘™šœ”š‘˜āˆ§šœ”š‘™+š‘ƒš‘–š‘—š‘˜š‘Žšœ”š‘˜āˆ§šœ”š‘Ž,(2.9) where š‘…š‘–š‘—š‘˜š‘™=āˆ’š‘…š‘–š‘—š‘™š‘˜,š‘ƒš‘–š‘—š‘˜š‘Ž=š‘ƒš‘–š‘˜š‘—š‘Ž. For the Landsberg curvature š‘ƒš‘–š‘—š‘˜āˆ¶=š‘ƒš‘–š‘›š‘—š‘˜, we have š‘ƒš‘–š‘—š‘˜=š›æš‘–š‘™š‘ƒš‘™š‘—š‘˜Ģ‡š“=āˆ’š‘–š‘—š‘˜,š‘ƒš‘›š‘—š‘˜=0.(2.10)

Similarly, the curvature 2-form of the Berwald connection š‘āˆ‡ can also be expressed as š‘‘š‘šœ”š‘–š‘—āˆ’š‘šœ”š‘˜š‘—āˆ§š‘šœ”š‘–š‘˜=š‘Ī©š‘–š‘—=12š‘š‘…š‘–š‘—š‘˜š‘™šœ”š‘˜āˆ§šœ”š‘™+š‘š‘ƒš‘–š‘—š‘˜š‘Žšœ”š‘˜āˆ§šœ”š‘Ž,(2.11) where š‘š‘…š‘–š‘—š‘˜š‘™=āˆ’š‘š‘…š‘–š‘—š‘™š‘˜,š‘š‘ƒš‘–š‘—š‘˜š‘Ž=š‘š‘ƒš‘–š‘˜š‘—š‘Ž.

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

Definition 2.1. For any š‘‹=š‘‹š‘–(šœ•/šœ•š‘„š‘–)āˆˆšœ‹āˆ—š‘‡š‘€, the Ricci curvature under the Berwald connection in the direction š‘‹ is given as š‘Ricš‘€1(š‘„,š‘¦,š‘‹)āˆ¶=āŸØš‘‹,š‘‹āŸ©š‘”ī“š‘—,š‘™,š‘ š›æš‘š‘˜š‘ š‘…š‘ š‘—š‘™š‘—(š‘„,š‘¦)š‘‹š‘˜š‘‹š‘™.(2.12) Obviously, if š‘‹=Ģ‚š‘’, then the Ricci curvature is just the common scalar Ricci curvature.

Definition 2.2. For any š‘‹,š‘Œāˆˆšœ‹āˆ—š‘‡š‘€, the directional section curvature of š‘€ under the Chern connection is given as š‘…š¾(š‘„,š‘¦,š‘‹āˆ§š‘Œ)=š‘–š‘—š‘˜š‘™(š‘„,š‘¦)š‘‹š‘–š‘Œš‘—š‘‹š‘˜š‘Œš‘™āŸØš‘‹,š‘‹āŸ©āŸØš‘Œ,š‘ŒāŸ©āˆ’āŸØš‘‹,š‘ŒāŸ©2.(2.13) In general, š¾(š‘„,š‘¦,š‘‹āˆ§š‘Œ)ā‰ š¾(š‘„,š‘¦,š‘Œāˆ§š‘‹). Particularly, if š‘€ is the Riemannian manifold, then š¾ is the Riemannian section curvature.

3. The Map between Finsler Manifolds

Let (š‘€,š¹) and (ī‚‹ī‚š‘€,š¹) be Finsler manifolds of dimension š‘› and š‘š, respectively, and let ī‚‹ī‚šœ™āˆ¶(š‘€,š¹)ā†’(š‘€,š¹) be a smooth map. š¹ and ī‚š¹ induce the metrics āˆ‘š‘”=š‘–(šœ”š‘–)2 and āˆ‘Ģƒš‘”=š›¼(ī‚šœ”š›¼)2, where {šœ”š‘–} and {ī‚šœ”š›¼} are the orthonormal one-form on š‘‡š‘€ and š‘‡ī‚‹š‘€, 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: š‘‘š‘‘š‘”šø(šœ™)š‘”āˆ£š‘”=0š‘›=āˆ’2š‘š‘›āˆ’1ī€œš‘†š‘€āŸØĢƒšœ,š‘‰āŸ©Ģƒš‘”š‘‘š‘‰š‘†š‘€,(3.1) where ī‚€ī‚āˆ‡Ģƒšœāˆ¶=š‘™ī‚1š‘‘šœ™(š‘™)=š¹2Ģƒšœš›¼šœ•šœ•Ģƒš‘„š›¼,Ģƒšœš›¼=šœ™š›¼š‘–š‘—š‘¦š‘–š‘¦š‘—āˆ’šœ™š›¼š‘˜šŗš‘˜+ī‚šŗš›¼,(3.2) where š‘™=š‘™š‘–(šœ•/šœ•š‘„š‘–),š‘™š‘–=š‘¦š‘–/š¹ is the dual field of the Hilbert form, and šŗš‘˜ and ī‚šŗš›¼ are the geodesic coefficients of (š‘€,š¹) and (ī‚‹ī‚š‘€,š¹), respectively. Here šœ™š›¼š‘–š‘—=šœ•2šœ™š›¼šœ•š‘„š‘–šœ•š‘„š‘—,šœ™š›¼š‘–=šœ•šœ™š›¼šœ•š‘„š‘–.(3.3) From the formula (3.1), we have

Lemma 3.1 (see [5]). Let šœ™ be harmonic map if and only if for any vector field š‘‰āˆˆš’ž(šœ™āˆ’1š‘‡ī‚‹š‘€), ī€œš‘†š‘€āŸØĢƒšœ,š‘‰āŸ©Ģƒš‘”š‘‘š‘‰š‘†š‘€=0.(3.4) ā€‰šœ™ is the strongly harmonic map if and only if Ģƒšœš›¼=0.

Let ī‚‹š‘€Ī¦āˆ¶š‘†š‘€ā†’š‘† be the map between the projective sphere bundles of š‘€ and ī‚‹š‘€, which is induced by šœ™. It is easy to find that Ī¦āˆ—āˆ˜š‘‘=š‘‘āˆ˜Ī¦āˆ—, šœ™āˆ—(ī‚šœ”š›¼)=šœ™š›¼š‘—šœ”š‘— is just the same as Ī¦āˆ—(ī‚šœ”š›¼)=šœ™š›¼š‘—šœ”š‘—. Let {šœ”š‘–,šœ”š‘Ž+š‘›} be the orthonormal frames of the dual bundle for š‘†š‘€, and let {ī‚šœ”š›¼,ī‚šœ”š”ž+š‘š} be the orthonormal frames of the dual bundle for š‘†ī‚‹š‘€. Then we have the following.

Proposition 3.2. Let ī‚‹š‘€Ī¦āˆ¶š‘†š‘€ā†’š‘† be the map between the projective sphere bundle of š‘€ and ī‚‹š‘€. Then Ī¦āˆ—ī€·ī‚šœ”š›¼ī€ø=šœ™š›¼š‘—šœ”š‘—,Ī¦āˆ—ī€·ī‚šœ”š›¼+š‘šī€ø=12Ģƒšœš›¼š‘—šœ”š‘—+šœ™š›¼š‘—šœ”š‘—+š‘›.(3.5) Obviously, if šœ™ is a strongly harmonic map, then Ī¦āˆ—(šœ”š›¼+š‘š)=šœ™š›¼š‘—šœ”š‘—+š‘›.

Proof. We will use natural frame to proof the Theorem. The relation between natural frame and moving frame satisfies šœ”š‘–=š¶š‘–š‘—š‘‘š‘„š‘—,šœ”š‘–+š‘›=š·š‘–š‘—š›æš‘¦š‘—,ī‚šœ”š›¼=š“š›¼š›½š‘‘š‘¢š›½,ī‚šœ”š›¼+š‘š=šµš›¼š›½š›æš‘£š›½,(3.6) where š“š›¼š›½,šµš›¼š›½,š¶š‘–š‘—,š·š‘–š‘— are orthonormal matrixes, and {š‘‘š‘„š‘–,š›æš‘¦š‘–} and {š‘‘š‘¢š‘–,š›æš‘£š‘–} are the natural bases of the dual bundle for š‘†š‘€ and š‘†ī‚‹š‘€, respectively. Then we have Ī¦āˆ—ī€·ī‚šœ”š›¼ī€ø=Ī¦āˆ—ī‚€š“š›¼š›½š‘‘š‘¢š›½ī‚=š“š›¼š›½Ī¦āˆ—ī€·š‘‘š‘¢š›½ī€ø=š“š›¼š›½š‘‘(š‘¢š›¼āˆ˜Ī¦)=š“š›¼š›½šœ•šœ™š›¼šœ•š‘„š‘–š‘‘š‘„š‘–=š“š›¼š›½šœ•šœ™š›¼šœ•š‘„š‘–ī€·š¶āˆ’1ī€øš‘–š‘—šœ”š‘—āˆ¶=šœ™š›¼š‘–šœ”š‘–,Ī¦āˆ—ī€·ī‚šœ”š›¼+š‘šī€ø=Ī¦āˆ—ī‚€šµš›¼š›½š›æš‘£š›½ī‚=šµš›¼š›½š‘‘(š‘£š›¼āˆ˜Ī¦)+šµš›¼š›½ī‚š‘š›½š›¾šœ•šœ™š›¾šœ•š‘„š‘—š‘‘š‘„š‘—=šµš›¼š›½ī‚µšœ•2šœ™š›¼šœ•š‘„š‘–šœ•š‘„š‘—š‘¦š‘–š‘‘š‘„š‘—+šœ•šœ™š›¼šœ•š‘„š‘—š‘‘š‘¦š‘—ī‚¶+šµš›¼š›½ī‚š‘š›½š›¾šœ•šœ™š›¾šœ•š‘„š‘—š‘‘š‘„š‘—=šµš›¼š›½ī‚µšœ•2šœ™š›¼šœ•š‘„š‘–šœ•š‘„š‘—š‘¦š‘–+ī‚š‘š›½š›¾šœ•šœ™š›¾šœ•š‘„š‘—ī‚¶ī€·š¶āˆ’1ī€øš‘—š‘˜šœ”š‘˜+šµš›¼š›½šœ•šœ™š›½šœ•š‘„š‘—ī€·š‘‘š‘¦š‘—+š‘š‘—š‘˜š‘‘š‘„š‘˜ī€øāˆ’šµš›¼š›½šœ•šœ™š›½šœ•š‘„š‘—š‘š‘—š‘˜š‘‘š‘„š‘˜=šµš›¼š›½ī‚µšœ•2šœ™š›¼šœ•š‘„š‘–šœ•š‘„š‘—š‘¦š‘–+ī‚š‘š›½š›¾šœ•šœ™š›¾šœ•š‘„š‘—āˆ’šœ•šœ™š›½šœ•š‘„š‘™š‘š‘™š‘—ī‚¶ī€·š¶āˆ’1ī€øš‘—š‘˜šœ”š‘˜+šµš›¼š›½šœ•šœ™š›½šœ•š‘„š‘™ī€·š·āˆ’1ī€øš‘™š‘—šœ”š‘—+š‘›1āˆ¶=2Ģƒšœš›¼š‘—šœ”š‘—+šœ™š›¼š‘—šœ”š‘—+š‘›.(3.7) So, we have completed the proof of the proposition.

4. The Rigidity Theorem

In the following, let (š‘€,š¹) be a Finsler manifold of dimension š‘›, and let (ī‚‹ī‚š‘€,š¹) be a Berwald manifold of dimension š‘š. Let ī‚‹ī‚šœ™āˆ¶(š‘€,š¹)ā†’(š‘€,š¹) 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 (š‘€,š¹) and (ī‚‹ī‚š‘€,š¹).

Let 1š‘’(šœ™)=2ī“š›¼,š‘–ī€·šœ™š›¼š‘–ī€ø2=12ā€–ā€–šœ™āˆ—ā€–ā€–2.(4.1) Define š‘šœ™š›¼š‘–|š‘—šœ”š‘—+š‘šœ™š›¼š‘–;š‘Žšœ”š‘Žāˆ¶=š‘‘šœ™š›¼š‘–āˆ’šœ™š›¼š‘—š‘šœ”š‘—š‘–+šœ™š›½š‘–Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€ø,(4.2)š‘šœ™š›¼š‘–|š‘—|š‘˜šœ”š‘˜+š‘šœ™š›¼š‘–|š‘—;š‘Žšœ”š‘Žāˆ¶=š‘‘š‘šœ™š›¼š‘–|š‘—āˆ’š‘šœ™š›¼š‘š‘™|š‘—šœ”š‘™š‘–āˆ’š‘šœ™š›¼š‘š‘–|š‘™šœ”š‘™š‘—+š‘šœ™š›½š‘–|š‘—Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€ø.(4.3) Differentiating Ī¦āˆ—ī‚šœ”š›¼=āˆ‘šœ™š›¼š‘–šœ”š‘– and by Ī¦āˆ—āˆ˜š‘‘=š‘‘āˆ˜Ī¦āˆ—, we have Ī¦āˆ—š‘‘ī‚šœ”š›¼=š‘‘šœ™š›¼š‘–āˆ§šœ”š‘–+šœ™š›¼š‘–š‘‘šœ”š‘–.(4.4) Substituting (2.8) and (4.2) into (4.4) yields Ī¦āˆ—ī€·š‘‘ī‚šœ”š›¼ī€ø=ī‚€š‘šœ™š›¼š‘–|š‘—šœ”š‘—+š‘šœ™š›¼š‘–;š‘Žšœ”š‘Ž+šœ™š›¼š‘—š‘šœ”š‘—š‘–āˆ’šœ™š›½š‘–Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī‚āˆ§šœ”š‘–āˆ’šœ™š›¼š‘–š‘šœ”š‘–š‘˜āˆ§šœ”š‘˜.(4.5) On the other hand, since Ī¦āˆ—ī€·š‘‘ī‚šœ”š›¼ī€ø=āˆ’Ī¦āˆ—ī‚€ī‚šœ”š›¼š›½āˆ§ī‚šœ”š›½ī‚=āˆ’Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øāˆ§Ī¦āˆ—ī‚€ī‚šœ”š›½ī‚=āˆ’Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øāˆ§šœ™š›½š‘–šœ”š‘–,(4.6) from (4.5), we have š‘šœ™š›¼š‘–|š‘—šœ”š‘—āˆ§šœ”š‘–+š‘šœ™š›¼š‘–;š‘Žšœ”š‘Žāˆ§šœ”š‘–=0.(4.7) That is, š‘šœ™š›¼š‘–;š‘Ž=0,š‘šœ™š›¼š‘–|š‘—=š‘šœ™š›¼š‘—|š‘–.(4.8) Differentiating (4.2) and from (4.8), we get š‘šœ™š›¼š‘–|š‘—š‘‘šœ”š‘—+ī‚€š‘‘š‘šœ™š›¼š‘–|š‘—ī‚āˆ§šœ”š‘—=āˆ’š‘‘šœ™š›¼š‘—āˆ§š‘šœ”š‘—š‘–āˆ’šœ™š›¼š‘—š‘‘š‘šœ”š‘—š‘–+š‘‘šœ™š›½š‘–āˆ§Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€ø+šœ™š›½š‘–Ī¦āˆ—ī€·š‘‘ī‚šœ”š›¼š›½ī€ø.(4.9) then by (2.8), (2.11), and (4.3), we have ī‚€š‘šœ™š›¼š‘–||š‘—||š‘˜šœ”š‘˜+š‘šœ™š›¼š‘–|š‘—;š‘Žšœ”š‘Ž+š‘šœ™š›¼š‘š‘™|š‘—šœ”š‘™š‘–+š‘šœ™š›¼š‘š‘–|š‘™šœ”š‘™š‘—āˆ’š‘šœ™š›½š‘–|š‘—Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī‚āˆ§šœ”š‘—=š‘šœ™š›¼š‘š‘–|š‘—šœ”š‘—š‘˜āˆ§šœ”š‘˜+ī‚€āˆ’š‘šœ™š›¼š‘—|š‘™šœ”š‘™āˆ’šœ™š›¼š‘™š‘šœ”š‘™š‘—+šœ™š›½š‘—Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī‚āˆ§š‘šœ”š‘—š‘–+šœ™š›¼š‘—ī€·š‘šœ”š‘—š‘˜āˆ§š‘šœ”š‘˜š‘–āˆ’š‘Ī©š‘—š‘–ī€ø+šœ™š›½š‘–Ī¦āˆ—ī‚€āˆ’ī‚šœ”š›¼š›¾āˆ§ī‚šœ”š›¾š›½+ī‚Ī©š›¼š›½ī‚+ī‚€š‘šœ™š›½š‘–|š‘—šœ”š‘—+šœ™š›½š‘™š‘šœ”š‘™š‘–āˆ’šœ™š›¾š‘–Ī¦āˆ—ī‚€ī‚šœ”š›½š›¾ī‚ī‚āˆ§Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€ø.(4.10) Simplifying (4.10) yields š‘šœ™š›¼š‘–|š‘—|š‘˜šœ”š‘˜āˆ§šœ”š‘—+š‘šœ™š›¼š‘–|š‘—;š‘Žšœ”š‘Žāˆ§šœ”š‘—=āˆ’šœ™š›¼š‘—š‘Ī©š‘—š‘–+šœ™š›½š‘–Ī¦āˆ—ī‚Ī©š›¼š›½.(4.11) Note that Ī¦āˆ—ī‚€ī‚Ī©š›¼š›½ī‚=Ī¦āˆ—ī‚€12ī‚š‘…š›¼š›½š›¾šœŽī‚šœ”š›¾āˆ§ī‚šœ”šœŽ+ī‚š‘ƒš›¼š›½š›¾š”žī‚šœ”š›¾āˆ§ī‚šœ”š”žī‚=12ī‚š‘…š›¼š›½š›¾šœŽšœ™š›¾š‘˜šœ™šœŽš‘˜šœ”š‘˜āˆ§šœ”š‘™+ī‚š‘ƒš›¼š›½š›¾š”žšœ™š›¾š‘˜šœ™š”žš‘šœ”š‘˜āˆ§šœ”š‘.(4.12) Substituting (2.11) and (4.12) into (4.11), by comparing the two sides of (4.11), we can get š‘šœ™š›¼š‘–|š‘—|š‘˜āˆ’š‘šœ™š›¼š‘–|š‘˜|š‘—=šœ™š›¼š‘™š‘š‘…š‘™š‘–š‘—š‘˜āˆ’ī‚š‘…š›¼š›½š›¾šœŽšœ™š›½š‘–šœ™š›¾š‘—šœ™šœŽš‘˜,š‘šœ™š›¼š‘–|š‘—;š‘Ž=šœ™š›¼š‘™š‘š‘ƒš‘™š‘–š‘—š‘Žāˆ’ī‚š‘ƒš›¼š›½š›¾š”žšœ™š›½š‘–šœ™š›¾š‘—šœ™š”žš‘Ž.(4.13)

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

Theorem 4.1. Let (š‘€,š¹) be a Finsler manifold, and let (ī‚‹ī‚š‘€,š¹) be a Berwald manifold. If the tension field of ī‚‹ī‚šœ™āˆ¶(š‘€,š¹)ā†’(š‘€,š¹) is zero, then ī“š‘—š’®š‘—āˆ£š‘—=||||āˆ‡š‘‘šœ™2+ī“š‘–,š‘—ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬š‘š‘…š‘–š‘—āˆ’ī“š‘–,š‘—ī‚€||šœ™āˆ—š‘’š‘–||2||šœ™āˆ—š‘’š‘—||2āˆ’ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—šœ™ī¬ī«āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬ī‚š¾ī‚‹š‘€ī€·šœ™āˆ—š‘’š‘–āˆ§šœ™āˆ—š‘’š‘—ī€ø,(4.14) where āˆ‘š’®=š›¼,š‘–,š‘—šœ™š›¼š‘–š‘šœ™š›¼š‘–|š‘—šœ”š‘—āˆ¶=š’®š‘—šœ”š‘—. In particular, if š‘€ and ī‚‹š‘€ are Riemannian manifolds, then āˆ‘š‘—š’®š‘—|š‘— is Ī”š‘’(šœ™).

Proof. Letting āˆ‘š‘’(šœ™)=1/2š›¼,š‘–(šœ™š›¼š‘–)2, then we have ī“de(šœ™)=š›¼,š‘–šœ™š›¼š‘–š‘‘šœ™š›¼š‘–.(4.15) Substituting (4.2) into (4.15) yields ī“de(šœ™)=š›¼,š‘–,š‘—ī‚€šœ™š›¼š‘–š‘šœ™š›¼š‘–|š‘—šœ”š‘—+šœ™š›¼š‘–šœ™š›¼š‘—š‘šœ”š‘—š‘–āˆ’šœ™š›¼š‘–šœ™š›½š‘–Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī‚.(4.16) By (2.8), we can get ī“š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—š‘šœ”š‘—š‘–=12ī“š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—ī‚€š‘šœ”š‘—š‘–+š‘šœ”š‘–š‘—ī‚ī“=āˆ’š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—š“š‘–š‘—š‘Žšœ”š‘Ž+ī“š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—Ģ‡š“š‘–š‘—š‘˜šœ”š‘˜.(4.17) From (2.6) and (3.5), we can obtain ī“š›¼,š›½,š‘–šœ™š›¼š‘–šœ™š›½š‘–Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€ø=12ī“š›¼,š›½,š‘–šœ™š›¼š‘–šœ™š›½š‘–Ī¦āˆ—ī‚€ī‚šœ”š›¼š›½+ī‚šœ”š›½š›¼ī‚ī“=āˆ’š›¼,š›½,š‘–ī‚š“š›¼š›½š”žšœ™š›¼š‘–šœ™š›½š‘–šœ™š”žš‘Žšœ”š‘Ž.(4.18) Substituting (4.17) and (4.18) into (4.16), we have ī“de(šœ™)=š›¼,š‘–,š‘—šœ™š›¼š‘–š‘šœ™š›¼š‘–|š‘—šœ”š‘—āˆ’ī“š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—š“š‘–š‘—š‘Žšœ”š‘Ž+ī“š›¼,š‘–,š‘—šœ™š›¼š‘–šœ™š›¼š‘—Ģ‡š“š‘–š‘—š‘˜šœ”š‘˜+ī“š›¼,š›½,š‘–ī‚š“š›¼š›½š”žšœ™š›¼š‘–šœ™š›½š‘–šœ™š”žš‘Žšœ”š‘Ž.(4.19) Defining āˆ‘š’®=š›¼,š‘–,š‘—šœ™š›¼š‘–š‘šœ™š›¼š‘–|š‘—šœ”š‘—āˆ¶=š’®š‘—šœ”š‘—, then we have ī“š‘—š’®š‘—āˆ£š‘—=ī“š‘–ī‚€š‘āˆ‡Ģ‚š‘’š‘–š’®š‘—šœ”š‘—ī‚ī€·Ģ‚š‘’š‘–ī€ø=ī“š‘–ī€·š‘‘š’®š‘–ī€øī€·Ģ‚š‘’š‘–ī€øāˆ’ī“š‘–,š‘—š’®š‘—š‘šœ”š‘—š‘–ī€·Ģ‚š‘’š‘–ī€ø=ī“š‘–ī€·š‘‘š’®š‘–ī€øī€·Ģ‚š‘’š‘–ī€øāˆ’ī“š‘–,š‘™,š‘—šœ™š›¼š‘™š‘šœ™š›¼š‘š‘™|š‘—šœ”š‘—š‘–ī€·Ģ‚š‘’š‘–ī€ø.(4.20) On the other hand, from (4.2) and (4.3), we have ī“š‘–ī€·š‘‘š’®š‘–ī€øī€·Ģ‚š‘’š‘–ī€ø=ī“š›¼,š‘–,š‘™š‘‘ī‚€šœ™š›¼š‘™š‘šœ™š›¼š‘™āˆ£š‘–ī‚ī€·Ģ‚š‘’š‘–ī€ø=ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘™|š‘–ī€·š‘‘šœ™š›¼š‘™ī€øī€·Ģ‚š‘’š‘–ī€ø+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘‘š‘šœ™š›¼š‘™|š‘–ī€·Ģ‚š‘’š‘–ī€ø=ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘™|š‘–ī‚€š‘šœ™š›¼š‘™|š‘–+šœ™š›¼š‘š‘šœ”š‘š‘™ī€·Ģ‚š‘’š‘–ī€øāˆ’šœ™š›½š‘™Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī€·Ģ‚š‘’š‘–ī€øī‚+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™ī‚€š‘šœ™š›¼š‘™|š‘–|š‘–+š‘šœ™š›¼š‘š‘™|š‘šœ”š‘š‘–ī€·Ģ‚š‘’š‘–ī€ø+š‘šœ™š›¼š‘š‘|š‘–šœ”š‘š‘™ī€·Ģ‚š‘’š‘–ī€øāˆ’š‘šœ™š›½š‘™|š‘–Ī¦āˆ—ī€·ī‚šœ”š›¼š›½ī€øī€·Ģ‚š‘’š‘–ī€øī‚=ī“š›¼,š‘–,š‘™ī‚€š‘šœ™š›¼š‘™|š‘–ī‚2+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘™|š‘–|š‘–+ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘™|š‘–šœ™š›¼š‘š‘šœ”š‘š‘™ī€·Ģ‚š‘’š‘–ī€ø+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘š‘|š‘–šœ”š‘š‘™ī€·Ģ‚š‘’š‘–ī€ø+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘š‘™|š‘šœ”š‘š‘–ī€·Ģ‚š‘’š‘–ī€ø.(4.21) Substituting (4.21) into (4.20) and by (2.8), we can get ī“š‘—š’®š‘–|š‘–=ī“š›¼,š‘–,š‘™ī‚€š‘šœ™š›¼š‘™|š‘–ī‚2+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘™|š‘–|š‘–+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘|š‘–ī€·š‘šœ”š‘š‘™+š‘šœ”š‘™š‘ī€øī€·Ģ‚š‘’š‘–ī€ø=ī“š›¼,š‘–,š‘™ī‚€š‘šœ™š›¼š‘™|š‘–ī‚2+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘™|š‘–|š‘–ī“+2š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘|š‘–Ģ‡š“š‘š‘™š‘–.(4.22) From (4.3), we have ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘™|š‘–|š‘–=ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘–|š‘–|š‘™+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™šœ™š›¼š‘š‘š‘…š‘š‘–š‘™š‘–āˆ’ī“š›¼,š‘–,š‘™ī‚š‘…š›¼š›½š›¾šœŽšœ™š›¼š‘™šœ™š›½š‘–šœ™š›¾š‘™šœ™šœŽš‘–.(4.23) Because šœ™ is a strongly harmonic map, from (3.2) and Lemma 3.1, we have that ī“š‘–š‘šœ™š›¼š‘–|š‘–=0,(4.24) Then ī“0=š‘–š‘‘š‘šœ™š›¼š‘–|š‘–ī€·Ģ‚š‘’š‘™ī€ø=ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘–|š‘–|š‘™+ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘š‘|š‘–šœ”š‘š‘–ī€·Ģ‚š‘’š‘™ī€ø+ī“š‘š›¼,š‘–,š‘™šœ™š›¼š‘š‘–|š‘šœ”š‘š‘–ī€·Ģ‚š‘’š‘™ī€ø.(4.25) So ī“š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘–|š‘–|š‘™ī“=āˆ’š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘|š‘–ī€·š‘šœ”š‘š‘–+š‘šœ”š‘–š‘ī€øī€·Ģ‚š‘’š‘™ī€øī“=āˆ’2š›¼,š‘–,š‘™šœ™š›¼š‘™š‘šœ™š›¼š‘|š‘–Ģ‡š“pil.(4.26) Substituting (4.23) and (4.26) into (4.22) yields ī“š‘—š’®š‘—|š‘—=ī“š›¼,š‘–,š‘™ī‚€š‘šœ™š›¼š‘™|š‘–ī‚2+ī“š›¼,š‘–,š‘™šœ™š›¼š‘™šœ™š›¼š‘š‘š‘…š‘š‘–š‘™š‘–āˆ’ī“š›¼,š‘–,š‘™ī‚š‘…š›¼š›½š›¾šœŽšœ™š›¼š‘™šœ™š›½š‘–šœ™š›¾š‘™šœ™šœŽš‘–.(4.27) Then we get Theorem 4.1.

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

Theorem 4.2. Let M be a compact Finsler manifold of dimension š‘›, and let ī‚‹š‘€ be a Berwald manifold of dimension š‘š. Suppose š‘Ž,š‘ are positive constants, for any š‘‹āˆˆšœ‹āˆ—š‘‡š‘€, š‘Ric(š‘‹)ā‰„š‘Ž, and š¾ī‚‹š‘€ā‰¤š‘, where š¾ī‚‹š‘€ is the directional section curvature of ī‚‹š‘€. Suppose the tension field of ī‚‹š‘€šœ™āˆ¶š‘€ā†’ is zero and šœ™ is nondegenerate. If š‘’š‘›(šœ™)ā‰¤š‘Ž2(š‘›āˆ’1)š‘,(4.28) then šœ™ is a constant map or totally geodesic map. In particular, if š‘’(šœ™)ā‰¤š‘Ž/2š‘, then šœ™ must be a constant map.

Proof. From Theorem 4.1, we have ī“š‘—š’®š‘—|š‘—=||||āˆ‡š‘‘šœ™2+ī“š‘–,š‘—ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬š‘š‘…š‘–š‘—āˆ’ī“š‘–,š‘—ī‚€||šœ™āˆ—š‘’š‘–||2||šœ™āˆ—š‘’š‘—||2āˆ’ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—šœ™ī¬ī«āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬ī‚š¾ī‚‹š‘€ī€·šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī€ø(4.29) Diagonalizing āŸØšœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—āŸ© at a point (š‘„,š‘¦)āˆˆš‘†š‘€, then we have ī«šœ™āˆ—š‘’š‘–,šœ™āˆ—š‘’š‘—ī¬=šœ†š‘–š›æš‘–š‘—.(4.30) Fixing a point (š‘„,š‘¦), then eigenvalues {šœ†š‘–} can be sorted as the following sequence: šœ†1(š‘„,š‘¦)ā‰„šœ†2(š‘„,š‘¦)ā‰„ā‹Æā‰„šœ†š‘›(š‘„,š‘¦)>0.(4.31) So, we have ī“š‘—š’®š‘—|š‘—ā‰„||||āˆ‡š‘‘šœ™2īƒ©āˆ’š‘4š‘’2(šœ™)āˆ’š‘›ī“š‘˜=1šœ†2š‘˜īƒŖ+2š‘Žš‘’(šœ™).(4.32) Then by the Schwarz inequality, we can get ī“š‘—š’®š‘—|š‘—ā‰„||||āˆ‡š‘‘šœ™2ī‚µ+2š‘’(šœ™)š‘Žāˆ’2(š‘›āˆ’1)š‘›ī‚¶š‘š‘’(šœ™).(4.33) Integrating the two sides of (4.33), we have ||||āˆ‡š‘‘šœ™2ī‚µ=0,(4.34)2š‘’(šœ™)š‘Žāˆ’2(š‘›āˆ’1)š‘›ī‚¶š‘š‘’(šœ™)=0.(4.35) If š‘’(šœ™)ā‰ 0, then šœ™ is a totally geodesic map and š‘’(šœ™)ā‰”(2(š‘›āˆ’1)/š‘›)š‘. In particular, if š‘’(šœ™)ā‰¤š‘Ž/2š‘, then š‘’(šœ™)<(š‘›/2(š‘›āˆ’1))(š‘Ž/š‘). So by (4.35), we have š‘’(šœ™)ā‰”0,(4.36) that is, šœ™ 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 thank anonymous referees for their valuable suggestions and pertinent criticisms.