Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).     

A class of Finsler metrics with isotropic S-curvature

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form. We characterize these metrics with isotropic S-curvature. Supported by a NNSFC grant (10671214) and by the NSF project of CQ CSTC. Supported by a NNSFC grant (10671214), a NSF grant (DMS-0810159) and the C. K. Chao Foundation for Advanced Research     

Einstein Finsler metrics and killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S ³ with Ric = 2F ², Ric = 0 and Ric = -2F ², respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

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.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

Principal bundles on compact complex manifolds with trivial tangent bundle

Let G be a connected complex Lie group and G ì G{Gamma subset G} a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E _H over G/Γ admits a holomorphic connection if and only if E _H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E _H over G/Γ admits a flat holomorphic connection if and only if E _H is homogeneous.     

On tangent bundles with Sasakian metrics of Finslerian and Riemannian manifolds

On the basis of the so-called phase completion the notion of vertical, horizontal and complete objects is defined in the tangent bundles over Finslerian and Riemannian manifold. Such a tangent bundle is made into a manifold of almost Kaehlerian structure by endowing it with Sasakian metric. The components of curvature tensors with respect to the adapted frame are presented. This having been done it is shown possible to study the differential geometry of Finslerian spaces by dealing with that of their own tangent bundles. This work was supported by National Research Coundil of Canada A-4037 (1960–70). Entrata in Redazione l'8 marzo 1970.     

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.     

On a class of weakly Einstein Finsler metrics

In this paper we study a special class of Finsler metrics—m-Kropina metrics which are defined by a Riemannian metric and a 1-form. We prove that a weakly Einstein m-Kropina metric must be Einsteinian. Further, we characterize Einstein m-Kropina metrics in very simple conditions under a suitable deformation, and obtain the local structures of m-Kropina metrics which are of constant flag curvature and locally projectively flat with constant flag curvature respectively.     

一类射影平坦的Finsler度量

本文主要研究由两个Riemann度量和一个1-形式构成的Finsler度量.首先,本文给出这类度量局部射影平坦的等价条件;其次,给出这类度量局部射影平坦且具有常旗曲率的分类情形;最后,构造这类度量局部射影平坦且具有常旗曲率K=-1的例子.     

On a class of projectively flat Finsler metrics

In this paper, we classify locally projectively flat general $(\alpha ,\beta )$-metrics $F=\alpha \varphi (b^2,\frac{\beta }{\alpha })$ on an $n(\ge 3)$-dimensional manifold if α is of constant sectional curvature and ${\varphi }_1\ne 0$. Furthermore, we find equations to characterize this class of metrics with constant flag curvature and determine their local structures.