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.

PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

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

Projectively flat exponential Finsler metric

In this paper, we study a class of Finsler metric in the form F=αexp(β/α)+εβ, where α is a Riemannian metric and β is a 1-form, ε is a constant. We call F exponential Finsler metric. We proved that exponential Finsler metric F is locally projectively flat if and only if α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of expo-nential Finsler metric F vanishes if and only if β is parallel with respect to α. And from this fact, we get that if exponential Finsler metric F is the Douglas metric, then F is not only a Berwald metric, but also a Landsberg metric.

Projectively flat arctangent Finsler metric

In this work, we study a class of special Finsler metrics F called arctangent Finsler metric, which is a special （α, β）-metric, where a is a Riemannian metric and β is a 1-form, We obtain a sufficient and necessary condition that F is locally projectively fiat if and only if α and β satisfy two special equations. Furthermore we give the non-trivial solutions for F to be locally projectively fiat. Moreover, we prove that such projectively fiat Finsler metrics with constant flag curvature must be locally Minkowskian.

On some projectively flat polynomial (α,β)-metrics

In this paper, we consider some polynomial （a,fl）-metrics, and discuss the sufficient and necessary conditions for a Finsler metric in the form F=α＋α1β＋α2β^2/α＋α4β^4/α^3 to be projectively flat, where ai 0=1,2,4） are constants with a1≠0, a is a Riemannian metric and β is a 1-form. By analyzing the geodesic coefficients and the divisibility of certain polynomials, we obtain that there are only five projectively flat cases for metrics of this type. This gives a classification for such kind of Finsler metrics.

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 a Class of Two-Dimensional Projectively Flat Finsler Metrics with Constant Flag Curvature

In this paper, we study a special class of two-dimensional Finsler metrics defined by a Riemannian metric and 1-form. We classify those which are locally projectively flat with constant 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 pro jectively flat.Furthermore,we classify locally pro jectively flat singular square metrics with constant flag curvature completely.

A simple remark on a flat projective morphism with a Calabi-Yau fiber

If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field,then any smooth connected fiber is also a K3 surface.Observing this,Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers.We shall give an explicit negative answer to his question in each dimension greater than or equal to three as well as a proof of his initial observation.

Projective Blaschke Manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds.

Ding w-Flat Modules and Dimensions

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.

The non-abelian Hodge correspondence on some non-K?hler manifolds

The non-abelian Hodge correspondence was established by Corlette(1988),Donaldson(1987),Hit chin(1987)and Simpson(1988,1992).It states that on a compact Kahler manifold(X,ω),there is a one-to-one correspondence between the moduli space of semi-simple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers.In this paper,we extend this correspondence to the projectively flat bundles over some non-Kahler manifold cases.Firstly,we prove an existence theorem of Poisson metrics on simple projectively flat bundles over compact Hermitian manifolds.As its application,we obtain a vanishing theorem of characteristic classes of projectively flat bundles.Secondly,on compact Hermitian manifolds which satisfy Gauduchon and astheno-K?hler conditions,we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle withΔ(E,?E)·[ωn-2]=0 must be an extension of stable Higgs bundles.Using the above results,over some compact non-Kahler manifolds(M,ω),we establish an equivalence of categories between the category of semi-stable(poly-stable)Higgs bundles(E,?E,φ)withΔ(E,?E)·[ωn-2]=0 and the category of(semi-simple)projectively flat bundles(E,D)with(-1)(1/2)FD=α■IdE for some real(1,1)-formα.

Gravitational Wave in Lorentz Violating Gravity

By making use of the weak gravitational field approximation, we obtain a linearized solution of the gravitational vacuum field equation in an anisotropic spacetime. The plane-wave solution and dispersion relation of gravitationaJ wave is presented explicitly. There is possibility that the speed of gravitational wave is larger than the speed of light and the easuality still holds. We show that the energy-momentum of gravitational wave in the ansiotropic spacetime is still well defined and conserved.