Finsler Geometry Of A Class Of Critical Metrics And Randers Metrics

The history of Finsler geometry can be traced back to the lecture given by B. Riemann in 1854, though he turned immediately to the quadratic case. P. Finsler was the first geometer who studied the general metrics systematically. In his thesis in 1918, Finsler developed a theory of curves and surfaces for such metrics. Therefrom, the name "Finlser space" was generally accepted. On the other hand, the correct notion of complex Finsler metrics was probably proposed firstly by G.B.Rizza ([52]). As important examples of complex Finsler metrics, the intrinsic metrics on complex manifolds play central roles in the studies of complex geometry.From 90's of the 20th century, encouraged by S.-S. Chern([25]), many geometersentered the field of Finsler geometry. In the last decade, the study of Finsler geometry has taken on a new look ([12], [58], [17], [60], [50], [24] and etc.). On the other hand, as a powerful tool, Finsler metrics are also widely applied to physics, biology, control theory, psychology and etc.([13])In this thesis, we will consider some properties of curvatures for Finsler metrics, both real and complex. The main text consists of three parts, which study Einstein-Hilbert functional, real Randers metrics and complex Randers metrics respectively.Einstein-Hilbert functionalLet M be an n-dimensional compact manifold. As is well known, among Riemannian metrics on M there is an important cla.ss of metrics called Einstein metrics, which are the critical points of the normalized Einstein-Hilbert functional where R_g is the scalar curvature of the Riemannian metric g, and dμ_M is the volume element of g. This motivates us to consider the corresponding functional in Finsler geometry. An attempt in this direction was tried by H.Akbar-Zadeh ([5]). Unfortunately, it seems that one could not obtain the tensor characteristic on generalized Einstein metrics from the variation calculus in [5] (also cf. D.Bao's comment [10]). It encourages us to look for the Finslerian analogue of critical metrics from the point of view of differential geometry and variational calculus.By virtue of the Chern connection on a Finsler manifold (M, F) with the Finsler metric F, we can define the flag curvature and the Ricci scalar, which are generalizations of the sectional curvature and the Ricci curvature in Riemanniangeometry, respectively ([14]). It is natural to define a similar functional in Finsler geometry by using the Ricci scalar and the volume form induced from the projective sphere bundle over (M,F). In fact, this functional can be defined bywhere Ric denotes the Ricci scalar and SM is the projective sphere bundle over M with volume element dμ_SM.One can check easily (0.2.2) is just the previous (0.2.1) if F is Riemannian by means of the integral trace formula or Lemma1.4 in [34].The purpose of the second chapter is to derive the Euler-Lagrange equation of the functional (0.2.2) and to study the properties of the critical metrics.The main difficulties in deriving the critical equation are a divergence formulaand a Green type formula. The former one was firstly obtained by H.AkbarZadeh,and was reproved by Xiaohuan Mo and Qun He-Yibing Shen. A special case of the Green type formula was firstly proved by Qun He-Yibing Shen, and we discover the following generic formLemma 0.2.1. Letψandφbe two smooth functions on SM. Then it holds where (g^ij) is the inverse of the fundamental tensor (g_ij) := (1/2[F²]_yⁱy^j.With the help of this lemma, we get the correct equation.Theorem 0.2.1. The Euler-Lagrange equation of the functional (0.2.2) iswherer=1/Vol(SM)∫_SMRic dμ_SM is the average of Ric on SM, J = J_idxⁱ is the meanLandsberg tensor, "|" and ";" denote respectively the horizontal and the vertical covariant derivatives with respect to the Chern connection, and "·" denotes the covariant derivative along the Hilbert form.Definition 0.2.1. A Finsler metric satisfying the equation (0.2.4) is called anε-critical metric.It is easy to show that a Riemannian metric is anε-critical metric iff it is Einstein. On the other hand, we have the following non-Riemannian examples.Example 0.2.1 Letαbe a Ricci-flat Riemannian metric andβbe parallel with respect toα, then the Randers metric F =α+βis anε-critical metric.Example 0.2.2 Let (M,g) and (N,h) be two Ricci-flat Riemannian manifolds, then the metricisε-critical on the product manifold M×N, where the functionφ(s, t) can be defined asHereεis a nonnegative real number and k is a positive integer.Applying some Ricci identities and the Hopf maximum principle, we obtain Theorem 0.2.2. Let M be a closed manifold, and F be a Finsler metric on M with positive constant flag curvature and constant S-curvature. Then F is anε-critical metric if and only if it is a Riemannian metric with positive constant sectional curvature.The S-curvature was discovered by Z.Shen in his study of volume comparison [58]. There are many rigidity results on the S-curvature. For instance, if F is reversible, then positive constant flag curvature and constant S-curvature will imply that F is Riemannian ([38]).From Theorem 0.2.2, one can see that even Finsler space forms might not beε-critical. So it is interesting to study the relation betweenε-critical metrics and Einstein-Finsler metrics. In dimension two, we haveProposition 0.2.1. On a closed surface, any nonpositively curved EinsteinFinslermetric isε-critical.Proposition 0.2.2. Let F be anε-critical metric on a surface. If its Landsberg scalar is horizontal constant, then F is an Einstein-Landsberg metric. Moreover, if the Ricci curvature is nonzero, then F is Riemannian.Real Randers metricsRanders metrics were introduced by G.Randers in the context of general relativity,^[51] and named by R.Ingarden in his paper on the electron microscope.^[36]Being the sum of a Riemannian metricα=(?) and a 1-formβ= b_i(x)yⁱ,Randers metrics are the favorite metrics for Finsler geometers, and are well understood by some great works such as [27], [17], [16], [64], [63] and etc.In Chapter 3. we first study theε-critical Randers metrics. By Zermelo navigation, [17] determined the Randers space forms. Particularly, any Randers metric with constant flag curvature must have constant S-curvature. So as a direct corollary of Theorem 0.2.2, we haveProposition 0.2.3. Anyε-critical Randers metric with positive constant flag curvature on a closed manifold must be a Riemannian metric with positive constantsectional curvature. In other words, the standard Riemannian metric on the sphere is isolated in the family of critical Randers metrics. On the other hand, Example 0.2.1 is nontrivial which belongs to the class of Berwald-Randers metrics. So it is interestingto understand what kind of Berwald-Randers metrics are indeed critical.The following proposition will answer this question.Proposition 0.2.4. A Berwald-Randers metric F =α+βisε-critical, if and only if it is one of the followings:(1)αis Einstein, andβ= 0;(2)αis Ricci-flat.Notice that the second class contains the Calabi-Yau manifolds, so it is ample. From another point of view, it gives a new significance of Calabi-Yau manifolds in Finsler geometry which says that Calabi-Yau manifolds will remain critical under certain perturbations.Besides the critical Randers metrics, Chapter 3 also considers Randers metricswith sectional flag curvature. The definition of sectional flag curvature is suggested in [62].A flag planted at a base point x on the manifold M, consists of a flagpole y∈T_xM and a sectionΠcontaining y. Typically, forΠspanned by {y, V}, the flag curvature of (y,Π) is K(x, y, V). Since it is independent of the choice of V inΠ, we also write it as K(x, y,Π).A Finsler metric is said to be of scalar flag curvature if the flag curvatureis independent of the section. In [48], it is verified that negatively curved Finsler metrics of scalar flag curvature on closed manifolds must be of Randers type(dim≥3). Later on, Randers metrics with scalar flag curvature were characterizedby [63]. Recently, [28] classified the Randers metrics with scalar curvature and constant S-curvature by Zermelo navigation(dim≥3).Now, what happens if the flag curvature is independent of the flagpole? Such metrics are called of sectional flag curvature since the flag curvature dependsonly on the section. Under this condition, the flag curvature can be written as K(x,Π) just as in Riemannian geometry. Clearly, Riemannian metrics are of sectional flag curvature, so this condition is non-Riemannian. It is natural to ask:Are there any nontrivial Finsler metrics with sectional flag curvature?As a start, we first study the Randers metrics as usually do. In dimensiontwo, sectional flag curvature means isotropic flag curvature. Since EinsteinRandersmetrics have been determined by [16], we may only consider high dimensions.Finally, we find it is a rigid condition for Randers metrics. Precisely, it saysTheorem 0.2.3. In dim≥3, any non-Riemannian Randers metric with sectional flag curvature must have constant flag curvature.Hence the characterization for Randers metrics is finished. Recently, some rigidity theorems for general Finsler metrics are verified in [30].Complex Randers metricsContrast to the real case, there are few examples in complex Finsler geometry except the Kobayashi metric and the Caratheodory metric. Following the spirit of real Randers metrics, N. Aldea and G. Munteanu began the study of complex Randers metrics ([7]) which have the formwhere a_ij(z)dzⁱ(?)dz^j is a Hermitian metric and b_i(z)dzⁱ is a (1,0)-form. In Chapter4, we mainly study the connection and holomorphic curvature of complex Randers metrics. Since all the complex Finsler metrics on Riemann surfaces are Hermitian, we will always assume that the complex dimension of the manifolds considered in this part is greater than one.In real Finsler geometry, a metric is said to be Berwaldian if it has linear Berwald connection. In 1981, Z.Szabo classified the Berwald metrics by noting that Berwald metrics share their holonomy groups with Riemannian metrics. In [2], the definition of complex Berwald space is given, and some conformal properties are verified.A complex Finsler metric is said to be Berwaldian if it is a Kahler-Finsler metric with linear Berwald connection. Here the notion of Kahler-Finsler metric is given by [8] where the Kahlerness is divided into three levels, i.e. strongly Kahler, Kahler and weakly Kahler. In Chapter 1, we prove that Kahler and strongly Kahler are in fact equivalent.Theorem 0.2.4. A Kahler-Finsler metric is actually strongly Kahler.Hence, there are only two Kahler conditions with respect to the ChernFinslerconnection in the complex Finsler geometry.As an analogue of real case, for complex Randers metrics we haveTheorem 0.2.5. A complex Randers metric F =α+ |β|is a Berwald metric if and only ifαis Kahlerian andβ(?)βis parallel with respect toα.Example 0.2.3 Let M be a usual Kahler manifold, and T be a flat torus. Choose a parallel (1,0)-formβon T, and letαbe the Kahlerian product metric on T×M. Then F =α+ |β| is a complex Berwald metric on T×M.Besides the connections, the holomorphic curvature is another important quantity in complex Finsler geometry. The holomorphic curvature K_F(z,v) is said to be isotropic, if it is independent of the direction v, i.e. K_F(z,v) = K_F(z). In Kahler geometry, isotropic holomorphic curvature implies certain rigidity. Then one may ask what happens in the complex Finsler realm? We study the complex Berwald-Randers metrics and partially answer this question. The main difficulty is to calculate the holomorphic curvature of Randers metrics. Fortunately,we obtain the formula and get a rigidity theorem.Theorem 0.2.6. A Berwald-Randers metric with isotropic holomorphic curvaturemust be either usually Kdhlerian or locally Mincowskian.Globally, we have Theorem 0.2.7. Let M be a simply connected complex manifold, and F =α+|β| be a complete Berwald-Randers metric on M. If F has isotropic holomorphic curvature, then it must be one of the following:(1) M = CPⁿ, F is the Fubini-Study metric;(2) M = {z∈Cⁿ : |z| < 1}, F is the Bergman metric;(3) M = Cⁿ,αis the Euclidean metric andβ(z,v) =β(0, v)e^iθ(z) whereθ(z) is a real function on M. Particularly, F is a Mincowskian metric.On the other hand, we say a complex Randers metric is strong ifαis Kahlerian andβis holomorphic. Then it holdsProposition 0.2.5. Let Mⁿ be a closed complex manifold with a strongly complex Randers metric F =α+ |β|. If the holomorphic curvature of F is positive, then F is a Hermitian metric.Here we give some examples which have different curvature properties.Example 0.2.4 On C²ⁿ, letα²=δ_ABdz^A(?)dz^B andβ=∑_i=1ⁿzⁱ⁺ⁿdzⁱ-zⁱdzⁱ⁺ⁿ.Then F(z,v)=α+|β| has zero holomorphic curvature.Example 0.2.5 On the unit discâ–³₁(?)Cⁿ,letαbe the Bergman metricandβbe a holomorphic one form with norm less than 1, for instanceβ=∑zⁱdzⁱ.Then the resulting Randers metric is complete and K_F≤-1/4.Example 0.2.6 Taking a small discâ–³_ε(?)Cⁿ in CPⁿ,letαbe the Fubini-Studymetric on itandβ= b_idzⁱ with constant b_i. Then K_F > 0 onâ–³_εfor sufficiently smallε.