PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

This article characterizes projectively fiat Finsler metrics with almost isotropic S-curvature.

S-curvature of isotropic Berwald metrics

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.

On Randers Metrics with Isotropic S-Curvature

In this paper, we study a class of Finsler metrics defined by a vector field on a Riemannian space form. We give an explicit formula for those with isotropic S-curvature. This class contains all Randers metrics of constant flag curvature.

On (α,β)-Metrics of Scalar Flag Curvature with Constant S-curvature

In this paper, we study （α,β）-metrics of scalar flag curvature on a manifold M of dimension n （n 〉 3）. Suppose that an （α,β）-metric F is not a Finsler metric of Randers type, that is, F ≠k1 V√α^2 ＋ k2β^2 ＋ k3β, where k1 〉 0, k2 and k3 are scalar functions on M. We prove that F is of scalar flag curvature and of vanishing S-curvature if metric. In this case, F is a locally Minkowski and only if the flag curvature K = 0 and F is a Berwald metric.

S-curvature of Doubly Warped Product of Finsler Manifolds

Doubly warped product of Finsler manifolds is useful in theoretical physics,particularly in general relativity.In this paper,we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.

On a class of two-dimensional Finsler manifolds of isotropic S-curvature

For an(α, β)-metric(non-Randers type) of isotropic S-curvature on an n-dimensional manifold with non-constant norm ‖β‖α, we first show that n = 2, and then we characterize such a class of two-dimensional(α, β)-manifolds with some PDEs, and also construct some examples for such a class.

Navigation Finsler metrics on a gradient Ricci soliton

In this paper,we study a class of Finsler metrics defined by a vector field on a gradient Ricci soliton.We obtain a necessary and sufficient condition for these Finsler metrics on a compact gradient Ricci soliton to be of isotropic S-curvature by establishing a new integral inequality.Then we determine the Ricci curvature of navigation Finsler metrics of isotropic S-curvature on a gradient Ricci soliton generalizing result only known in the case when such soliton is of Einstein type.As its application,we obtain the Ricci curvature of all navigation Finsler metrics of isotropic S-curvature on Gaussian shrinking soliton.

NATURALLY REDUCTIVE(α_(1),α_(2))METRICS

Letting F be a homogeneous(α_(1),α_(2))metric on the reductive homogeneous manifold G/H,we first characterize the natural reductiveness of F as a local f-product between naturally reductive Riemannian metrics.Second,we prove the equivalence among several properties of F for its mean Berwald curvature and S-curvature.Finally,we find an explicit flag curvature formula for G/H when F is naturally reductive.

ON(α,β)-METRICS OF CONSTANT FLAG CURVATURE

In this paper,we study the(α,β)-metrics of constant flag curvature.We characterize almost regular(α,β)-metrics of constant flag curvature under the condition that β is a homothetic 1-form with respect to a.Furthermore,we prove that if a regular(α,β)-metric is of constant flag curvature and β is a Killing 1-form with constant length,then it must be a Riemannian metric or locally Minkowskian.

Properties of Berwald scalar curvature

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature.As a consequence,Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.For(α,β)-metrics on manifold of dimension greater than 2,if the mean Landsberg curvature and the Berwald scalar curvature both vanish,then the Berwald curvature also vanishes.

Eigenvalue Comparison Theorems on Finsler Manifolds

Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas-Weyl （α,β）-metrics. Suppose that a regular （α,β）-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.

On a Class of Weakly-Berwald （α,β）-Metrics

In this paper, we study an important class of (α,β)-metrics in the form F = (α + β)m+1/αm on an n-dimensional manifold and get the conditions for such metrics to be weakly-Berwald metrics, where α = aij(x)yiyj is a Riemannian metric and β = bi(x)yi is a 1-form and m is a real number with m = 1,0,1/n. Furthermore, we also prove that this kind of (α,β)-metrics is of isotropic mean Berwald curvature if and only if it is of isotropic S-curvature. In this case, S-curvature vanishes and the metric is weakly-Berwald metric.

The Characterizations and Constructions of Sprays of Isotropic Curvature

This paper discusses the sprays of isotropic curvature.We first determine the relationship betweenχ-curvatures of two projectively related sprays.Based on this,we find an approach to construct sprays of isotropic curvature and find infinitely many sprays of isotropic curvature via some known sprays of isotropic curvature.In particular,by using famous Funk metricΘ,we can construct infinitely many sprays of isotropic curvature,some of which can be induced by Finsler metrics,but others cannot be induced by any Finsler metric.

On conformal complex Finsler metrics

In this paper,we study conformal transformations in complex Finsler geometry.We first prove that two weakly Kahler Finsler metrics cannot be conformal.Moreover,we give a necessary and sufficient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal weakly Kahler Finsler.Finally,we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric,which preserve the geodesics,holomorphic S-curvatures and mean Landsberg tensors.

Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below

We obtain the Laplacian comparison theorem and the Bishop- Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ri∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.