Some Problems In Finsler And Sasakian Geometry

In this paper, we will study several problems of Finsler and Sasaki geometry.First of all, we will study a special kind of Finsler metric, the （α,β）-metric. We focus on the Killing vector fields on the manifolds equipped with this metric. The maximum dimension of the killing vector space in a non-Riemannian （α,β）-space is given. correspondingly, the non-Riemannian （α,β）-metric which admits the maximum dimension of killing vector space is also determined. Further more, we study the gap phenomenon by assuming a is a homogenious Riemannian metric. The first gap is given. The examples of different dimensions of killing vector space in low dimensional case can be determined. It shows the gap phenomenon almost disappear in the （α,β） space.Secondly, we will study the Randers metric. We will mainly discuss the con-formally Einstein problem. The condition of the conformal factor ρ such that the Randers metric being conformally Einstein is given. Those equivalents can be sim-plified once we noticed the Einstein condition of Randers metric. Based on that, we claim the Randers metric with s0≠0can not be conformal to an Einstein met-ric nontrivially. A Randers metric is conformally Einstein must has conformally isometric S-curvature. If added some conditions, we can get rigidity theorem.Thirdly, For the complex Finsler manifold, we notice a description of the holo-morphic curvature. Using this definition, we get the Schwarz lemma between com-plex Finsler manifolds and the Hartogs phenomenon of the complex Finsler mani-fold.On the Sasaki geometry, firstly, we notice the （Φ,J）-holomorphic map is basic. This irritate us to establish the Schwarz lemma of such map. In this procession, we get a Bochner type inequality. As an application of this inequality, we study the curvature properties and the existence of the （Φ,J）-holomorphic maps on Sasakian manifolds. We also find a basic homotopy invariant of the （Φ, J）-holomorphic maps. it can be seen this invariant only holds in the basic homotopy class. Another important aspect of the Sasakian geometry is the Sasaki-Ricci flow. We can get the poincare type inequality and other lemmas. We also proved the energy of the transverse Ricci potential tends to zero along the Sasaki-Ricci flow. Based on the other people’s work, we extend the Perelman’s non-collapsing theorem. By this theorem, we get the gradient estimation of the transverse Ricci potential.As the consists of the modern differential geometry, both Finsler and Sasaki ge-ometry are play important roles and get applications in Maths and Physics. Though they can be connected in some way, but they can have much connection from Maths or Physics purpose. It also an interesting problem itself. 0.3Part1:Finsler geometryIn1854, Riemann first introduced the concept of metric. Since then, the geom-etry is changed a lot. Though he mention the metrics in the general case, Riemann still focused on the metrics of quadratic forms, for the complexity of computation. This metric is the famous Riemannian metric. In1918, Finsler first studied in his doctoral dissertation the metrics without the restriction of being quadratic and the curves and surfaces in such spaces [41]. This metric has been named after him as the Finsler metric [30].Definition. A Finsler structure on a manifold M is a nonnegative function F:TM→[0,∞) with the following properties:（a）F is C∞in the entire slit tangent bundle T M\0;（b）F（x, λy）=λF（x,y） for any λ>0;（c）The Hessian matrix （gij）:=1/2[（F2）yiyj] is positive-definite at any point of T M\0.In the past two centuries, Geometrists are really interested in Riemannian geometry. Meanwhile, various quantities in Finsler geometry are found [36],[35], [87],[52],[75]. It’s not until Professor S. S. Chern’s promotion in1990s that the Finsler geometry has a golden period. There are two important aspects. One is to generalize the results from Riemannian geometry [96],[79],[94],[53]),[97],[54],[24],[9]. Another one is to discuss the relationships of the non-Riemannian quantities [4],[10][98],[25],[85],[68]. The discussion and application of them are needed badly [102]. In addition to those, since it is a more general metrc geometry than the Riemannian one, the application of Finsler geometry are related with Physics, Psychology, information, biology and so on [1],[4],[5],[11].As a special kind of Finsler metric,（α,β）-metric are studied by lots of mathe-matic workers for it is a computable Finsler metric [77],[95],[33].The （α,β）-metric is important and it is consisted of a Riemannian metric α and a1-form β.Definition. A Finsler metric is called （α, β）-metric, if it can be expressed as F=αφ（s） where s=α-β and φ is α C∞positive function on an open interval （-bo,bo）.φ satisfies φ（s）-sφ’（s）+（b2-s2）φ"（s）>0,|s|≤b<bo.It is easy to see that if φ=/1+Cs2, where C is a constant, then F is Riemannian. If φ=1+s, F=α+β which is called Randers metric which is firstly studied in1941by G.Randers. A lot, of works about （α,β）-metric promote the development of Finsler geometry. With such metrics, a lot of interesting works are found [6].Killing vectors are the vector fields on a manifold that preserve the metric. They are also the infinitesimal generators of isometrics. They get important ap-plications in both physics and geometry. It’s known very early that the dimension of linear space formed by killing vectors can be at most n（n+l）/2in n dimen-sional Riemannian space. Hsien Chung Wang proved that any Finsler manifold admits the most parameters of group of motions must be the Riemannian manifold with constant curvature [111]. This rigidity result can comprehended as one way of the description of the Finsler manifolds by isomeric group. In the late20th cen-tury and early21th century, physicists consider this problem again by some new approaching[65],[66]. They propose some new theories to adjust Einstein’s relativ-ity, for example DSR [19],[58] and VSR [31]. The former assumes a new energy constant and the latter refhies the group of motions. Those theories are related with Finsler geometry in some way [46],[45],[78]. Based on this fact, the research of Killing vector fields on （α, β）-space are important. We first give the complete proof of the equations of the Killing vector fields on （α,β）-space, which is first given by X. Li Z. Chang and X. Mo in [65]. And more, we can ensure the maximum dimension of killing vector space of an （α,β）-space is n（n-1）/2+1. conversely, we get the following theorem.Theorem0.1. A non-Riemannian （α,β）-space M admits the maximum isometries if and only if the metric α is α product of α constant curvature Rie-mannian metric and a1-dimensional flat metric ds2+dt2, i.e （M, α） is a product manifold of a constant curvature manifold M and a1-dim flat space R β is a parallel vector on R. In this case,（M,α） is a product manifold of a Riemannian manifold M with constant curvature and R.Irritated by Hsien Chung Wang’s work [111], there were lots of work about the isometry groups generated by Killing vector fields in Riemannian spaces [39]. The most famous one is the book iThc differential geometry on the homogeneous Riemannian spaces? written by C. Gu [120]. As in the Riemannian case, suppose α being a homogeneous Riemannian metric, the （α,β）-spaces also admit gap phe-nomenon. Here we use the first and the second gaps to describe the cases.Theorem0.2. Let M be an n-dimensional （α, β）-space. The first gap of M is from n（n+1）/2to n（n-1）/2+1. The second gap is from n（n-1）/2to （n-1）（n-2）/2+3, when n≥8.The gap phenomenon almost disappear in the non-Riemannian （α,β）-spaces, especially in the lower dimensional case. We just consider the4and5dimensional case to illustrate this situation, since the Lie subalgebra of o（n） is a pure algebra problem.In1941, physicist Randers introduced a non-Riemannian metric to describe the relativity [84], It was named after him, called Randers metric. Randers metric is a very important Finsler metric since it’s form is concise and background is clearly. As a more special （α,β）-metric, it provides a mass of examples to research Finsler geometry [26],[100],[27],[25]. In2002, D. Bao etc. used the navigation to discuss the Randers metrics [7],[99], Given such metrics a new physical background. Other-wise, Randers metrics has unexpected interpretation on the asymmetrical universe space [23],[67]. In1960, Yamabe conjecture any compact Riemannian manifold is conformal to a Riemannian manifold of constant scalar curvature [112]. This ques-tion was raised to the famous Poincare conjecture. By the unremitting work of lots mathematicians, R. Schoen finally proved Yamabe conjecture in1984[92]. There is no scalar curvature in Finsler geometry. Accordingly, the Ricci scalar plays an important role. So we may ask the question as the Yamabe problem in Finsler case: Is a Finsler manifold can conformal to an Einstein manifold? The researches of Einstein matric are always important and urgent in both mathematics and Physics. D. Bao etc. has find the equivalent condition of a Randers metric to be Einstein [8]. The Einstein conditions and properties on special spaces are also be studied [115],[28]. We first focus on the conditions of a Randers metric to be conformal to an Einstein metric. We have the equivalent conditions of the conformal factor. From those conditions, we can get the following rigidity theorem.Theorem0.3. If a Randers metric with s0≠0is conformally Einstein, the conformal transformation must be homothetic.If further assume the1-form β is a Killing vector field of a, we can get the following rigidity theorem.Theorem0.4. Let F=α+β be a Randers metric with β is a Killing vector about α. If F is conformal Einstin, the conformal transformation must be homothetic. Further more, if the Ricci curvature of a along β is nonpositive, then F is conformally Einstein if and only if a is Ricci flat. In this case, the metric is Berwald.The study of complex Finsler metric can trace back to the study of Caratheodory metric. The famous Caratheodory metric and Kobayashi metric are both the strongly pseudoconvex comples metrics on the strongly convex domain [64]. On the general complex manifold, it’s much easy to find a Finsler metric than a Her-mitian metric [63]. It’s meaningful to study complex Finsler geometry for complex analysis, analytic functions of several complex variables, vector bundle theory and algebra geometry. On this aspect, J. Cao, Pit-Mann Wong, K. Chandler, T. Aikou etc. have contributed a lot [3],[22],[21]. Schwarz lemma is a basic and important theorem in complex analysis. Ahlfors first introduce geometric concept, the Guass curvature, into Schwarz lemma [2]. In1978, Yau proved the Schwarz lemma for the holomorphic maps from complete Kahler manifolds to compact Hermitian manifolds [113]. Since then, many mathematicians, especially the Chinese mathematicians, got plentiful results in this area [118],[119],[107]. In this article, we prove thatTheorem0.5. Let （M1, F1）,（M2,F2）be two Finsler manifolds and（M1,M2） is compact. Their corresponding holomorphic curvatures satisfy KF1≥-B,KF2≤-A for A, B>0. Let φ be the holomorphic map from M1to M2. Then φ*F22≤B/AF21.The Hartogs phenomenon is from the Hartogs extension theorem. The theorem say that, when n≥2and0≤a <b, any holomorphic function defined in a spherical shell Dna,b={z∈Cn|a<|z|2<b} can be extended to the ball Bnb（of radius centered a the origin）. In the other words, there exists a holomorphic function on Bb whose restriction on Dna,b is just f. We call a manifold M obeys Hartogs theorem, for m≥2and any0≤a<b, any holomorphic map from Dma,b into M can be extended to a holomorphic map from Bmb into M [49]. Griffiths and Shiffman proved any complete Hermitian manifold with non-positive holomorphic sectional curvature obeys the Hartogs phenomenon [48],[104]. Based on the proof of them, we prove thatTheorem0.6. Any complete complex Finsler manifold with non-positive holomorphic curvature obeys the Hartogs phenomenon.0.4Part2:Sasaki geometrySasaki geometry can be considered as the dual of Kahler geometry on dimen-sion. In1962, Sasaki in article [88] first to study the manifolds endowed with regular contact metric structure. Such structure is known as the Sasaki structure. A man-ifold with such structure is called Sasakian manifold. From1990’s, some European mathematicians looked into this geometry and got vast important works [?],[12],[13],[14],[15]. AdS/CFT dual are meaningful in both theoretical physics and math-ematics. In1999, Klebanov and Witten discussed an interesting generalization of such dual [70],[62]. It is refer to a cone Y of dimension6with Ricci fiat. If pre-serving the N=1supersymmetry, Y must be a Calabi-Yau manifold, whose level hypersurface is a Sasaki-Einstein manifold. Based on such requirement of quantum field theory and conformal field theory, many physicists and mathematicians have had abundant discussions in Sasaki-Einstein metric and Sasaki-Ricci flow [106],[50],[51],[71]. For the Sasakian manifold, except the early works [89],[90],[91], C.P. Boyer, K. Galicki, J. Kollor, P. Matzeu also looked into the structures of it [16],[17],[72],[73],[34]. and made the classification. Futaki etc. also generalized the Futaki tensor [42]. R. Petit proved the strongly rigidity theorem, which belong to Y. Siu [105], in [82] by using the property of Dirac’s operator.As the dual of Kahler manifolds, the geometry structures on Sasakian manifold are abundant. Without the metric g, the Reeb field ξ is a Killing vector field. Any function satisfies ξf=df（ξ）=0are called basic. The dual of ξ can define a global1-form η. Also as the complex structure in Kahler geometry, there is a （1,1） type tensor Φ on any Sasakian manifold.The holomoorphic maps on complex manifolds are first discussed by J.Eells and L.Lemaire as a special case of the harmonic maps [37],[38]. After that, Y. Siu [105], S. Yau [122], Y. Xin [121] ctc. also studied it. We first consider the （Φ, J）、 holomorphic maps on Sasakian manifolds. Noticing such maps are basic, we can proveTheorem0.7. Let （M,ξ,η,Φ,g） be a complete Sasakian manifold with Ricci curvature bounded from below by-K1, and （N, H. J） be a Hermitian manifold with holomorphic bisectional curvature bounded from above by-K2, where K1;K2are constants, and K1≥2, K2>0. Then for any±（Φ,J）-holomorphic mapping f from M to N, we have If K1<2and K2>0, then any±（Φ, J）-holomorphic map f from M to N must be trivial. As an application of the Bochner’s inequality got in the proof of Schwartz lemma, we use it to get the following theorem.Theorem0.8. Let （M,g,ξ,η,Φ） be a complete noncompact Sasakian man-ifold without boundary of dimension2m+1. Let R（x） denote the pointwise lower bound of the transverse Ricci curvature of M and R_x is negative part of R（x）. Assume R_x satisfies for som.e p> m, and someβ<m2, where Br（y） denote the ball centered at y with radius r. Suppose f be a non-constant （Φ, J）-holomorphic map from M to a Herrnitian manifold （N, h, J） which has holomorphic bisectional curvature bounded from above by K（z） for all z E N. Suppose that the curvature of the image of M under f satisfies K（f（x））<-B for some constant B>0and for all x M. Then it must satisfy the inequality In particular, if∫M RdV>0, then f has to be identically constant.As a special kind of harmonic maps, the holomorphic maps has homotopy invariants. This result can be found in the topics of harmonic maps of J.Eells and L.Lemaire [38]. In this article, we attempt to extend such result into the Sasakian case. We find a basic homotopy invariant on the Sasakian manifolds. Further more, this invariant are only preserved in the basic homotopy class.Theorem0.9. Let （M,g,ξ,η,Φ） be a compact Sasakian manifold and （N,h,J） be a Kdhler manifold. Suppose f be a （Φ,J）-holomorphic map from M to N. Let E’f), E"（f） be the integral of e （f）=|df-J o df o Φ|2and e"’（f）=|df+J o df o Φ|2respectively. Then E’（f）-E"（f） is an invariant in its basic homotopic class. Moreover, the invariant is just hold in the class of basic maps.Like the history stated above, Sasaki-Ricci flow is a main point of the study of Sasaki manifolds. The short time and long time existence are proved by G. Wang etc. in [106]. They also got the existence of the soliton when the basic first Chern class arc positive. P. Guan and X. Zhang used the Mongc-Ampere equation to study the transverse Sasaki-Ricci flow [51].On the Sasaki manifold, looking for the Einstein metric need the study of Sasaki-Ricci flow. When the metric convergent to an Einstein metric? In what sense for this convergence? The results in the nonpositive curvature case have been got. So the positive case are more important. To answer this question, it require the estimations of the gradient of transverse Ricci potential and the curvature. Using the Poincare inequality and the Perelman’s non-collapsing theorem, one can get the estimating of the average of transverse Ricci potential. Considering such convergence, in this case, we first prove the gradient and the curvature are small when the potential is small itself by using the extremum principle.Theorem0.10. There are positive constant only depending on dimension n such that for any∈with0<∈<δ and any time to>0, if||h（to）||co≤∈, thenOn the other hand, As in the Kahler case, supposing the Mabuchi κ-energy are bounded below, we can prove the transverse Ricci potential energy is tends to zero along the Sasaki-Ricci flow.Theorem0.11. The Mabuchi K-energy is bounded below on the πc1B （M）. Set Y（t）=∫M|▽h|2（dη）m▽η/=||▽η||2L2be the energy of the Ricci potential. Then for any∈πcR1（M）, Y（t）→0along the Sasaki-Ricci flow as t→∞.