On conformal complex Finsler metrics

In this paper, we study conformal transformations in complex Finsler geometry. We first prove that two weakly Kähler 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 Kähler Finsler. Finally, we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric, which preserve the geodesics, holomorphic S curvatures and mean Landsberg tensors.

     

Projective complex Finsler metrics

In this paper we obtain the conditions under which two complex Finsler metrics are projective, i.e. have the same geodesics as point sets. Two important classes of such metrics are considered: conformal projective and weakly projective complex Finsler spaces. For each of them we study the transformations of the canonical connection. We pay attention to local projectivity in a pure Hermitian or Kähler space.     

Projectively related complex Finsler metrics

In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák’s theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert’s Fourth Problem is also pointed out. As an application, the projectiveness of a complex Randers metric is described.     

On dually flat Finsler metrics

In this paper, we study the locally dually flat Finsler metrics which arise from information geometry. An equivalent condition of locally dually flat Finsler metrics is given. We find a new method to construct locally dually flat Finsler metrics by using a projectively flat Finsler metric under the condition that the projective factor is also a Finsler metric. Finally, we find that many known Finsler metrics are locally dually flat Finsler metrics determined by some projectively flat Finsler metrics.     

Some remarks on projectively flat complex Finsler metrics

In this paper, we hall discuss the projective flatness of complex Finsler metrics by investigating the geometry of projective bundles associated with a holomorphic vector bundle.     

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature ?4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature ?4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.     

On a new class of Finsler metrics

In this paper, the geometric meaning of (α,β)-norms is made clear. On this basis, a new class of Finsler metrics called general (α,β)-metrics are introduced, which are defined by a Riemannian metric and a 1-form. These metrics not only generalize (α,β)-metrics naturally, but also include some metrics structured by R. Bryant. The spray coefficients formula of some kinds of general (α,β)-metrics is given and the projective flatness is also discussed.     

On a class of Douglas Finsler metrics

In this article, we study a class of Finsler metrics called general(α, β)-metrics,which are defined by a Riemannian metric α and a 1-form β. We determine all of Douglas general(α, β)-metrics on a manifold of dimension n 2.     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

Finsler metrics with constant (or scalar) flag curvature

By finding Killing vector fields of general Bryant??s metric we give a lot of new Finsler metrics of constant (or scalar) flag curvature and determine their scalar curvature.     

Finsler metrics with constant （or scalar） flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K =3c + σ where σ and c are scalar functions on M.In this paper,we establish the intrinsic re...     

On a class of projectively Ricci-flat Finsler metrics

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover, we construct some nontrivial examples on such Finsler metrics.