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.

Minimal Surfaces and Gauss Curvature of Conoid in Fins-ler Spaces with (<i>α</i>, <i>β</i>)-Metrics

In this paper, minimal submanifolds in Finsler spaces with (α, β)-metrics are studied. Especially, helicoids are also minimal in (α, β)-Minkowski spaces. Then the minimal surfaces of conoid in Finsler spaces with (α, β)-metrics are given. Last, the Gauss curvature of the conoid in the 3-dimension Randers-Minkowski space is studied.

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.

UNIQUE COMMON FIXED POINT OF A FAMILY OF SELF-MAPS WITH SAME TYPE CONTRACTIVE CONDITION IN 2-METRIC SPACE

In this paper, we prove that a family of self-maps {Ti,j}i,j∈N in 2-metric space has a unique common fixed point if （i） {Ti,j}i,j∈N satisfies the same type contractive condition for each j ∈ N; （ii） Tm,μ .Tn,v = Tn,v.Tm.μ for all m,n,μ,v ∈ N with μ≠v. Our main result generalizes and improves many known unique common fixed point theorems in 2-metric spaces.

New Unique Common Fixed Point Results for Four Mappings with Ф-Contractive Type in 2-Metric Spaces

In this paper, some new unique common fixed points for four mappings satisfying Ф-contractive conditions on non-complete 2-metric spaces are obtained, in which the mappings do not satisfy continuity and commutation. The main results generalize and improve many well-known and corresponding conclusions in the literatures.

Uniqueness of Common Fixed Points for a Family of Mappings with <i>Φ</i>-Contractive Condition in 2-Metric Spaces

In this paper, we will introduce a class of 5-dimensional functions Φ and prove that a family of self-mappings {Ti,j} iεN in 2-metric space have an unique common fixed point if 1) {Ti,j} iεN satisfies Φj-contractive condition, where ΦjεΦ, for each jεN;2) Tm,μ n,v for all m,n,μ,vεN with μ ≠ v . Our main result generalizes and unifies many known unique common fixed point theorems in 2-metric spaces.

Four Mappings Satisfying Ψ-Contractive Type Condition and Having Unique Common Fixed Point on 2-Metric Spaces

In this paper, we introduce a class Ψ of real functions defined on the set of non-negative real numbers, and obtain a new unique common fixed point theorem for four mappings satisfying Ψ-contractive condition on a non-complete 2-metric space and give the versions of the corresponding result for two and three mappings.

An Improvement of a Known Unique Common Fixed Point Result for Four Mappings on 2-Metric Spaces

In this paper, we introduce a new class Γ, which is weak than a known class Ψ, of real continuous functions defined on [0, +∞), and use another method to prove the known unique common fixed point theorem for four mappings with γ-contractive condition instead of Ψ-contractive condition on 2-metric spaces.

Unique Common Fixed Points for Two Weakly C~*-contractive Mappings on Partially Ordered 2-metric Spaces

In this paper, we give existence theorems of common fixed points for two mappings with a weakly C*-contractive condition on partially ordered 2-metric spaces and give a sufficient condition under which there exists a unique common fixed point.

Common Fixed Points for Two Mappings with Implicit-linear Contractions on Partially Ordered 2-metric Spaces

In this paper, we introduce a new class U of 3-dimensional real functions, use U and a 2-dimensional real function ? to construct a new implicit-linear contractive condition and obtain some existence theorems of common fixed points for two mappings on partially ordered 2-metric spaces and give a sufficient condition under which there exists a unique common fixed point. The obtained results goodly generalize and improve the corresponding conclusions in references.

Common Fixed Point Theorems for Generalized Cyclic Contraction Pairs in b2-metric Spaces

In this paper,we introduce the notion of generalized cyclic contraction pairs in b2-metric spaces and establish some fixed point theorems for such pairs.Then,we give an example to illustrate our results.

Projective Changes between Generalized (<i>α</i>, <i>β</i>)-Metric and Randers Metric

Projective change between two Finsler metrics arises from Information Geom-etry. Such metrics have special geometric properties and will play an important role in Finsler geometry. The purpose of the present paper is to find a relation to characterize the projective change between generalized (α, β) - metric ( μ1, μ2 and μ3 ≠ 0 are constants) and Randers metric , where α and are two Riemannian metrics, β and are 1-forms. Further, we study such projective change when generalized (α, β) -metric F has some curvature property.

Projectively Ricci-flat General(α,β)-metrics

In this paper,we study the projectively Ricci-flat general(α,β)-metrics within to a spray framework and also bring out the rich variety of behaviour displayed by an important projective invariant.Projective Ricci curvature is one of the essential projective invariant in Finsler geometry which has been introduced by Z.Shen.The projective Ricci curvature is defined as Ricci curvature of a projective spray associated with a given spray G on M^(n) with a volume form dV on M^(n).

On Conformally Flat(α,β)-metrics with Special Curvature Properties

In this paper, we study a significant non-Riemannian quantity E-curvature, which is defined by S-curvature. Firstly, we obtain the formula of E-curvature for （α,β）-metrics. Based on it, we show that the E-curvature vanishes for a class of （α,β）-metrics. In the end, we get the relation of E-curvature for conformally related Finsler metrics, and classify conformally flat （α,β）-metries with almost vanishing E-curvature.

Projectively flat Asanov Finsler metric

In this work, we study the Asanov Finsler metric F=α（β^2/α^2＋gβ/α＋1）^1/2exp{（G/2）arctan[β/（hα）＋G/2]}, where α=（αijy^iy^i）^1/2 is a Riemannian metric and β=by^i is a 1-fom, g∈（-2,2）, h=（1-g^2/4）^1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and ,Sis parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.

On Douglas General(α,β)-metrics

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general （α, β）-metrics, which are defined by a Riemannian metric and a . We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.

α-相对抛物仿射球的Bernstein性质

设y:M→An+1是一个局部严格凸的超曲面,由定义在一个凸域ΩAn上的严格凸函数xn+1=f(x1,x2,…,xn)给出.考虑M上的α-相对度量Gα=ρα∑2fxixjdxidxj,我们研究关于度量Gα完备的α-相对抛物仿射球M得到:若其α-Ricci曲率有下界,则M一定是椭圆抛物面.

Some projectively flat (α,β)-metrics

In this paper, we study a class of Finsler metrics in the form , where is a Riemannian metric, form, and ∈ and k≠0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.

PROJECTIVELY FLAT MATSUMOTO METRIC AND ITS APPROXIMATION

Abstract In this article, the author studies the projectively flat Matsumoto metric F=α^2/（α -β）, where α=√αijy^iy^j is a Riemannian metric and β =biy^i is 1-form. Theyconclude that α is locally projectively fiat and β is paralled with respect to α. And get the same result for the higher order approximate Matsumoto metric.

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.