Finsler Manifolds, Harmonic Maps And Projective Flat Asanov Measure

The content of this paper is divided into three parts corresponding to three chapters. In the first chapter, we study the harmonic maps from a complex Finsler manifold to a Hermitian manifold. By the first and second variation formulasof (?)- energy, we obtain some existence theorems and homotopy invariant theorems. In the second chapter, we give the upper bound estimation for energy density of totally geodesic maps between real Finsler manifolds. By this estimationwe can get the generalize Schwarz lemma from real Finsler manifold to Riemannian manifold. In the third chapter, we give the sufficient and necessary conditions for Asanov Finsler metric to be projectively fiat and to be a Douglas metric.As S.S.Chern, the international famous geometrist, said that Finsler metrics are just Riemannian metrics without quadratic restriction[13], which was firstly introduced by B.Riemann in 1854. Finsler geometry is to study the geometric properties of a manifold with Finsler metrics. Recently, under the encouragement of S.S.Chern, studies on Finsler geometry have taken on a new look[11][44][45]. On the other hand, as a powerful tool, Finsler geometry can be also applied to biology, physics, control theory, psychology and etc[1] [2] [3] [4] [8]. Finsler geometry has become an important branch of Riemannian geometry.·Harmonic maps for complex Finsler manifoldsComplex geometry is an important part of differential geometry. The quantumphysics especially, has stimulated the study of complex structures. G.Rizza[41] provided, through an analogy with the real case, the definition of complex Finsler metrics. Recently, there has been an increased interest in complex Finsler metrics[1] [5] [6] [12].In[26], Kobayashi gives two good reasons for considering complex Finsler structures. One is that every hyperbolic complex manifold M carries a natural complex Finsler metric in a broad sense. The second reason is as differential geometric tool in the study of complex vector bundles because in [27], Kobayashi proved that a holomorphic vector bundle E over a compact complex manifold is negative, that is, its dual vector bundle E^* is ample, if and only if it admitsa strongly pseudoconvex Finsler metric F with negative curvature. Since then, complex Finsler metrics have attracted much renewed attention to possible applications in various fields in complex geometry.Harmonic maps are important and interesting in both differential geometry and mathematical physics. Some open problems in Finsler geometry have been proposed in [35]. One of the open problems is to study harmonic maps between Finsler manifolds. By using the volume measure induced from the projective sphere bundle, harmonic maps between real Finsler manifolds were investigated in [22] [23] [33] [34] [42].Harmonic maps on complex Finsler manifold is also an interesting subject in the study of geometry. Recently, Nishikawa studied the harmonic maps from a compact Riemann surface into complex Finsler manifolds via considering the (?)-energy [38]. When the target manifold is complex Finsler manifold, the energy density function will be any serious singularity, then we only to study harmonic maps from complex Finsler manifolds to Hermitian manifolds; in particular, to Kahler manifolds, by virtue of the (?)-energy.Let (M, G) be a complex manifold of dimension m with strongly pseudoconvexFinsler metrics G, and (N, H) be a Hermitian manifold of complex dimensionn. Letφ: M→N be a smooth map from M to N. Then we havewhere T^1,0M is the holomorphic tangent bundle of M. Then the (?)-energy density ofφcan be defined naturally bywhereWe now consider a smooth variation ofφ=φ₀ via a family of smooth maps Let V₀:=dφ_t((?)/(?)t)_t=0 be the variation vector field ofφ, then by the complicate calculation, we can get the first variation formulas of (?)-energy function:Theorem 0.1. Let (M,G) be a compact complex Finsler manifold and (N,H) be a Hermitian manifold. Letφ: M→N be a smooth map from M to N. Then the first variation of (?)-energy functional iswhereIf (M, G) is a compact strongly Kahler Finsler manifold, then we have:Corollary 0.2. Let (M, G) be a compact strongly Kahler Finsler manifold and (N, H) be a Hermitian manifold. Letφ: M→N be a smooth map from M to N. Then the first variation of (?)-energy functional iswhere(?)It is well known that a harmonic map is critical point of the first variation of the energy functional. Note that V₀(z) is the variation field on M, Q^α(z, v) is the function on PM. In order to give the definition of harmonic maps, letDefinition A.φis harmonic if and only if‖Q‖≡0.φis called strongly harmonic if and only if Q^α= 0.From this definition, we see that, a holomorphic (resp. anti-holomorphic) map is a (strongly) harmonic map. Also we remark that the strong harmonicity implies the harmonicity. One basic problem for harmonic maps is to study the existence of harmonic maps. In [34] an existence theorem of harmonic maps from a real Finsler manifold to a Riemnnian manifold was given. On the other hand, in order to study the harmonic maps from a Hermitian manifold M into a Riemannian manifold N, J.Jost and Yau introduced and studied a nonlinear elliptic system in [25]. In local coordinate, the system iswhereγ_αβ is the Hermitian metric of M,Γ_jkⁱ are the Christoffel symbols of N, andα,β,…= 1,…, dimM, i, j,…= 1,…, dimN. The maps which satisfying (1) are called Hermitian harmonic maps. In general, Hermitian harmonic is not harmonic unless M is Kahler. By means of the study on the solutions of the system (1), some existence theorems were given in [25].In the following, let (M, G) be a compact strongly Kahler Finsler manifold and (N, H) be a compact Kahler manifold. Letψ: M→N be a smooth map from M to N. From Definition A,φis a harmonic map if and only iffor any variation field V₀. The invariant volume form of PM iswherethen (2) can be changed intowhereγ^ij(z):=∫_PzMG^ij(z,v)det(G_kl(z,v))dσ/∫_PzMdet(G_kl(z,v))dσ,σ(z):=∫_PzMdet(G_kl[(z,v))dσ.Since thefield V₀ is arbitrary on M, thenφis a harmonic map if and only if Letφ^α=f^α+(?)f^n+α,then (3) becomewhere 1≤A, B, C,…≤2n. Comparing (1) with (4) and using the existenceresults due to J.Jost and S.T.Yau in [25], we have immediately the following:Theorem 0.3. Let (M,G) be a compact strongly Kahler Finsler manifold and (N,H) be a compact Kahler manifold with negative sectional curvature. Letψ: M→N be continuous, and suppose thatψis not homotopic to a map onto a closed geodesic of N. Then there exists a harmonic mapφ: M→N homotopic toψ.Theorem 0.4. Let (M,G) be a compact strongly Kahler Finsler manifold, (N, H) be a compact Kahler manifold with nonpositive sectional curvature. Letψ: M→N be smooth andε(g^*TN)≠0, where E is the Euler class. Then there exists a harmonic map f homotopic toψ.Remark. We can also get the other existence theorems as in [25]. By means of the volume measure of the projective tangent bundle PM, we can define the (?)-energy and (?)-energy ofφrespectively bywhere c_M is the standard volume of the (m -1)-dimensional complex projective space CP^m-1. We also have E(φ) = E'(φ)+ E"(φ).Obviously,φis holomorphic (resp. antiholomorphic) if and only if E_?= 0 (resp. E_? = 0). Setby comparing the first variation formula of (?)-energy and (?)-energy, we can getThis proves the following:Theorem 0.5. If (M, G) is a compact strongly Kahler Finsler manifold, (N, H) is a usual Kahler manifold. Then K(φ) is a smooth homotopy invariant; i.e., it is constant on the connected components of C(M,N).Remark. When (M, G) is Kahler manifold, this striking theorem due to A.Lichnerowicz ([31]).By Theorem 0.5, we can get the following results:Corollary 0.6. Ifφis a holomorphic (antiholomorphic) map from a compact strongly Kahler Finsler manifold to a Kahler manifold, then it is a harmonic map and an absolute minimum of E in its homotopy class.Corollary 0.7. Ifφ₀ andφ₁ are homotopy maps from a compact strongly Kahler Finsler manifold to a Kahler manifold withφ₀ holomorphic andφ₁ antiholomorphic,thenφ₀ andφ₁ are constant. In particular, any homotopically trivial holomorphic(antiholomorphic) map is constant.By the second variation formula ofφ, we can get the following stable theorem:Theorem 0.8. Let (M, G) be compact complex Finsler manifold and (N, H) be a curvature flat Kahler manifold. Then any harmonic map from (M, G) to (N,H) is stable.Remark. Recently, B.Chen and professor Yibing Shen prove that Kahler and strongly Kahler Finsler metrics are in fact eqivalent ([54]). So, all the strongly Kahler Finsler manifolds in this chapter can be changed into Kahler Finsler manifolds.·The upper bound estimation of energy density between real Finsler manifoldsIn the second section of the thesis, we give the upper bound estimation for energy density of totally geodesic maps between real Finsler manifolds. By this estimation we can get the generalize Schwarz lemma from real Finsler manifold to Riemannian manifold.Let (M, F) and (M, F) are real Finsler manifolds of dimension n and m respectively.φ: (M, F)→(M, F) is non-degenerate smooth map. Suppose {xⁱ,yⁱ} be the local coordinates on TM, (x, [y]) is the point of projectively tan- gent bundle SM, where [y]={λy :λ> 0} is the ray through y and starting from x.Let TM = TM \ {0}, then the canonical projectionÏ€: TM→M gives rise to a convector bundleÏ€^*T^*M→SM which has a global sectionω:= (?)F/(?)yⁱdxⁱ called Hilbert form, whose dual vector field is l =yⁱ/F(?)/(?)xⁱ=lⁱ(?)/(?)xⁱ,viewed as a global section of the vector bundleÏ€^*TM. {e_i} is an othonormal basis ofÏ€^*TM→SM with respect to g = g_ijdxⁱ (?) dx^j, where g_ij = 1/2(F²)_yⁱy^j.It is well-known that there exists uniquely the Chern connection ^câ–½onÏ€^*TM→SM,ω_jⁱ is the connection 1-form. The curvature 2-forms of Chern connection ^câ–½arewhere R_{j kl}ⁱ=-R_{j lk}ⁱ,P_{j ka}ⁱ=P_{k ja}ⁱ,{ω^j,ω^n+a} is an orthonormal basis on thedual bundle of SM, andFirstly, several related definitions are given in the following:Definition B.For any X=Xⁱ(?)/(?)xⁱ∈π^*TM,the Ricci curvature underChern connection in the direction X is given asObviously, if X = e, then the Ricci curvature is just the common scalar Ricci curvatur.Definition C. For any X,Y∈π^*TM,the directional section curvature ofM under Chern connection is given asIn general, K(x,y,X∧Y)≠K(x,y,Y∧X). Particularly, if M is Riemannian manifold, then K is Riemannian section curvature.Under what condition we can give the estimation of the energy density is an interesting problem in Riemannian and Finsler geometry. In particular, if the map is harmonic map the this is the famous Schwarz lemma. Many results about the generalizations of Schwarz lemma have been obtained [16] [19] [20] [48] [51].Definition D. Let(M,g) and (M,g) are real Finsler manifolds,φbe a smooth map from M to M.φ^*g = B_ijω^ωj, where (B_ij) = (∑_αφ_i^αφ_j^α) be the matrix of n×n, the elements of the matrix B(x,y) are differentiable functions on SM. Letλ₁(x,y)≥λ₂(x,y)≥…≥λ_n(x,y) > 0 denote the eigenvalues ofB(x,y). If there exists a positive constant k≥1,satisfying (?)/(?)≤k, then we sayφis a k-dilatation map.In particularly, if (M, g) and (M, g) are Riemannian manifolds, the definition of k-dilatation is given in [19]. In [19], S.I.Goldberg,T.Ishihara, and N.C.Petridis got the generalization of the lemma for harmonic mappings of k-dilatation.In the second chapter, we give the upper bound estimation for totally geodesic maps between two real Finsler manifolds. Specifically, we can get:Theorem 0.9. Let (M, F) be a compact Finsler manifold and (M, F) be a Finsler manifold. Letφ: (M, F)→(M, F) be a non-degenerate totally geodesic mapping of k-dilatation with vanishing tension field and (x_o,y_o)∈SM is maximumpoint of‖φ_*‖, where‖φ_*‖= (?), e(φ) be the energy density ofφ. If the Cartan tensor and Carton form on M satisfies |A|≤C₁ , |▽η|≤C₁,|▽η|≤C₁ on the point (x_o,y_o). Also, Ricci curvature ^CRic^M(X)≥- C₃ for all X∈π^*TM and the directional sectional curvature of M satisfying K^M≤-C₄, the Cartan tensor of M is C¹ bounded and C₂ is the upper bound, thenwhere C_i, i = 1,2,3 are nonnegative constant, C₄ is a positive constant and satisfyingC₂ < 2C₄/p²k², and |·| denotes the length with repect to the Riemannian metric g which induced from the Sasaki metric on SM.If the target manifold is Riemannian, then the condition of totally geodesic in Theorem 0.9 can be changed into harmonic maps. By Theorem 0.9 we can obtain:Proposition 0.10. Let (M, F) be a compact Finsler manifold and (M, F) be a Riemannian manifold. Letφ: (M, F)→(M, F) be a non-degenerate harmonic mapping of k-dilatation with vanishing tension field and (x_o, y_o)∈SM is maximumpoint of‖φ_*‖,where‖φ_*‖=(?),e(φ) be the energy density ofφ. If theCartan tensor and Cartan form on M satisfies |A|≤C₁,|▽η|≤C₁,|▽η|≤C₁on the point (x_o, y_o). Also, Ricci curvature ^CRic^M(X)≥-C₃ for all X∈π^*TMand the Riemannian sectional curvature of M satisfying K^M≤-C₄,thenwhere C₁, C₃ are nonnegative constant, C₄ is a positive constant. p = rankφ_*(x,y)≥2,|·| denotes the length with repeat to the Riemannian metric g which inducedfrom the Sasaki metric on SM.φis distance decreasing if C₄≥1/2p²k²(C₃+2C₁).Proposition 0.10 is just the generalize Schwarz lemma from Finsler manifold to Riemannian manifold, which has been obtained by Weiping Du in his doctor thesis [55]. When (M,F) is also a Riemannian manifold, C₁ will also be zero, then we get the result in [19].·Protectively flat Asanov metricThis is the Hilbert's 4th problem to characterize the (not-necessarily-reversible) distance functions on an open subset in Rⁿsuch that straight lines are geodesics[24]. Regular distance functions with straight geodesics are projectively flat Finsler metrics. Projectively flat Finsler metrics have been studied by many mathematicians [29] [36] [40] [43] [47] [52] etc.The metrics which they study are all (α,β)-metrics, which can be expressed aswhere a =(?) is a Riemannian metric,β= b_iyⁱ is a one-form.φ=φ(s) is a C^∞positive function on an open interval (-b₀, b₀) satisfyingIt is known that F is a Finsler metric if and only if‖β_x‖_α0 for any x∈M.Forsome Finsler metrics, F =αφ(β/α) is projectively flat if and only ifαis projectively flat andβis parallel with respect toα, the such projectively flat (α,β)-metrics are called to be trivial.Recently, B. Li and Z. Shen [30] study and classified the projectively (α,β)-metrics with constant flag curvature.In [4], G. S. Asanov introduce a sigular special Finsler metric in the Minkowski spaces, which analogous to Randers metric. It can be expressed in the following form:where g∈(-2,2),h=(?),G=g/h.When g = 0, F is a Riemannianmetric.It is easy to find that the functionφ(s) satisfying (6), so F is really a (α,β)-metric. We call F the Asanove Finsler metric. For Asanov metric, we shall firstly prove the following:Theorem 0.11. Asanov metric is locally projectively flat if and only if the following conditions holds:(a)βis parallel with respect toα,(b)αis locally projectively flat. That is,αis of constant curvature.By this theorem, it is easy to observe that the projectively flat Asanov metric is trivial.A theorem due to Douglas states that a Finsler metric F is projectively flat if and only if two special curvature tensors are zero. The first is the Douglas tensor. The second is the projective Weyl tensor for n≥3, and the Berwald-Weyl tensor for n = 2. It is know that the projective Weyl tensor vanishes if and only if the flag curvature of F have no dependence on the transverse edges (but can possibly depend on the position x and the flagpole y). If the Douglas tensor of F vanish, we call F is a Douglas metric.S. Bacso and M. Matsumoto [7] proved that a Randers metric F =α+βis a Douglas metric if and only ifβis a closed 1-form. M. Matsumoto [32] has proved that when n = dimM≥3, F =α²+β²/αis a Douglas metric if and only if whereÏ„=Ï„(x) is a scalar function. For Asanov metric, we shall also prove thefollowing:Theorem 0.12. The Douglas tensor of Asanov metric vanishes if and only ifβis parallel with respect toα.