Matsumoto Metrics of Constant Flag Curvature: A Puny Class of Finsler Metrics with Constant Curvature

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a closed n-dimensional manifold of dimension n ≥ 3 is either Riemannian or locally Minkowskian.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

一类新的具有标量旗曲率Finsler度量

In this paper,we study a new class of general(α,β)-metrics F defined by a Riemannian metric α,a 1-form β and C ∞ function φ(b2,s).We provide the projective factor of a class of general(α,β)-metrics F=αφ(b2,s),and apply these formulae to compute its flag curvature.     

Projectively Flat Singular Square Metrics with Constant Flag Curvature

In this paper, we study and characterize locally projectively flat singular square metrics with constant flag curvature. First, we obtain the sufficient and necessary conditions that singular square metrics are locally projectively flat. Furthermore, we classify locally projectively flat singular square metrics with constant flag curvature completely.     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

On the Dynamics of Uniform Finsler Manifolds of Negative Flag Curvature

The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.     

Constant Scalar Curvature Metrics on Toric Surfaces

The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist. Received: May 2008, Revision: December 2008, Accepted: December 2008     

两类具有标量旗曲率的弱Berwald度量

In this paper, we study the （α,β）-metrics of scalar flag curvature in the form of F = α ＋ εβ ＋ κβ^2/α （ε and k ≠ 0 are constants） and F = α^2/α-β. We prove that these two kinds of metrics are weak Berwaldian if and only if they are Berwaldian and their flag curvatures vanish. In this case, the metrics are locally Minkowskian.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式,给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致.此外,通过推导射影球丛纤维上的散度公式,给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式。给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致．此外,通过推导射影球丛纤维上的散度公式。给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果．     

On a New Class of Projectively Flat Finsler Metrics

A class of Finsler metrics with three parameters is constructed. Moreover, a sufficient and necessary condition for this Finsler metrics to be projectively flat was obtained.     

具有可反Douglas曲率的Matsumoto度量

本文得到Matsumoto度量具有可反Douglas曲率的充分必要条件,该条件蕴含存在具有可反Douglas曲率的非Douglas的Finsler度量.     

On Conformal Metrics with Constant Q-Curvature

We review some recent results in the literature concerning existence of conformal metrics with constant Q-curvature.The problem is rather similar to the classical Yamabe problem:however辻 is characterized by a fourth-order operator that might lack in general a maximum principle.For several years existence of geometrically admissible solutions was known only in particular cases.Recently;there has been instead progress in this direction for some general classes of conformal metrics.     

ON THE METRICS OF THE RIEMANNIAN MANIFOLDS WHICH ADMIT ISOMETRIC IMBEDDING INTO SPACE OF ANY CONSTANT CURVATURE

In this paper the term“Riemannian manifold”means that the fundamental quadraticdifferential form may be indefinite.     

A Remark on Kahler Metrics of Constant Scalar Curvature on Ruled Complex Surfaces

In this paper we point out how some recent developments in thetheory of constant scalar curvature Kähler metrics canbe used to clarify the existence issue for such metrics in thespecial case of (geometrically) ruled complex surfaces. 2000Mathematics Subject Classification 53C55, 58E11.     

On Discrete Constant Mean Curvature Surfaces

Recently, a curvature theory for polyhedral surfaces has been established that associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gaussian image face. Therefore, a study of constant mean curvature (cmc) surfaces requires studying pairs of polygons with some constant nonvanishing value of the discrete mean curvature for all faces. We focus on meshes where all faces are planar quadrilaterals or planar hexagons. We show an incidence geometric characterization of a pair of parallel quadrilaterals having a discrete mean curvature value of ?1. This characterization yields an integrability condition for a mesh being a Gaussian image mesh of a discrete cmc surface. Thus, we can use these geometric results for the construction of discrete cmc surfaces. In the special case where all faces have a circumcircle, we establish a discrete Weierstrass-type representation for discrete cmc surfaces.     

On Higher Eta-Invariants and Metrics of Positive Scalar Curvature

Let N be a closed connected spin manifold admitting one metric ofpositive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving ₁(N) and dim N, for N to admit an infinite number of metrics of positive scalar curvature that are nonbordant. Mathematics Subject Classifications (2000) 55N22, 19L41.     

一类射影平坦芬斯勒度量

In this paper, the authors study a class of Finsler metric defined by a Rieman- nian metric and a 1-form. We find a necessary and sufficient condition for the metric to be prejectively flat.