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1.
There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before. 相似文献
2.
We classify the almost regular weakly stretch non-Randers-type(α, β)-metrics with vanishing Scurvature. In the class of regular metrics, they reduce to Berwald ones. Here, we demonstrate that when an almost regular weakly stretch non-Randers-type(α, β)-metric with vanishing S-curvature is not Berwaldian, then it is a weakly generalized unicorn. This yields an extension of Zou-Cheng and Chen-Liu's theorems. Finally, we show that any projective non-Randers β-change of a unicorn is a unicorn. 相似文献
3.
Under the assumption that' is a strongly convex weakly Khler 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. 相似文献
4.
In this paper, we study an important class of (α,β)-metrics in the form F = (α+β)^m+1/α^m on an n-dimensional manifold and get the conditions for such metrics to be weakly- Berwald metrics, where α = √aij(x)y^iy^j is a Riemannian metric and β = bi(x)y^i 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. 相似文献
5.
《数学学报(英文版)》2017,(7)
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 metricα=(a_(ij)(x)y~iy~j)~(1/2) and a 1-form β= b_i(x)y~i. 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. 相似文献
6.
We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric. 相似文献
7.
Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation. 相似文献
8.
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. 相似文献
9.
Finsler空间上的Weyl曲率 总被引:1,自引:0,他引:1
MoXiaohuan 《高校应用数学学报(英文版)》2005,20(1):10-20
The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type. 相似文献
10.
朱红梅 《数学物理学报(B辑英文版)》2018,38(2):695-708
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. 相似文献
11.
in this psper,we investigate nore general rings than GPP-rings,called fPP-rings.First,we in-vestigate fPP-rings and their classical quotient quotient rings.We ptove (1) fPP-rings are f-quasi-regular rings.(2)R is a fPP-ring then Q(R) is fPP-ring.(3)R= iRi is a fPP-ring if and only if every Ri is a fPP-ring.Second,we present a characterization of fPP-ring via fP-injectivity,we prove that R is a fPP-ring if and only if every quotient module of a imjective R-module is fP-injectiv if and only ifevery quotient module of a P-injective R-module is fP-injective.Third,we study how fPP-rings are related to von Neu-mann regular rings,we prove that R is von Nevmann regular if and only if R is fPP-ring and for every α∈R,there is b∈E(R) and d∈R suth that α=f(α)b and f(α)=f^2(α) d for some f∈F(R).Finally,we give a example of fPP-ring which is not GPP-ring. 相似文献
12.
Letting F be a homogeneous(α1, α2) metric on the reductive homogeneous manifold G/H, we first characterize the natural reductiveness of F as a local f-product between naturally reductive Riemannian metrics. Second, we prove the equivalence among several properties of F for its mean Berwald curvature and S-curvature. Finally, we find an explicit flag curvature formula for G/H when F is naturally reductive. 相似文献
13.
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). 相似文献
14.
The authors extend the notion of statistical structure from Riemannian geometry to the general framework of path spaces endowed with a nonlinear connection and a generalized metric. Two particular cases of statistical data are defined. The existence and uniqueness of a nonlinear connection corresponding to these classes is proved. Two Koszul tensors are introduced in accordance with the Riemannian approach. As applications, the authors treat the Finslerian (α, β)-metrics and the Beil metrics used in relativity and field theories while the support Riemannian metric is the Fisher-Rao metric of a statistical model. 相似文献
15.
Let X be a non-elementary Riemann surface of type(g,n),where g is the number of genus and n is the number of punctures with 3g-3+n1.Let T(X)be the Teichmller space of X.By constructing a certain subset E of T(X),we show that the convex hull of E with respect to the Teichmller metric,the Carathodory metric and the Weil-Petersson metric is not in any thick part of the Teichmler space,respectively.This implies that convex hulls of thick part of Teichmller space with respect to these metrics are not always in thick part of Teichmller space,as well as the facts that thick part of Teichmller space is not always convex with respect to these metrics. 相似文献
16.
In this article, after giving a necessary and sufficient condition for two Einstein- Weyl manifolds to be in conformal correspondence, we prove that any conformal mapping between such manifolds is generalized concircular if and only if the covector field of the conformal mapping is locally a gradient. Using this fact we deduce that any conformal mapping between two isotropic Weyl manifolds is a generalized concircular mapping. Moreover, it is shown that a generalized concircularly flat Weyl manifold is generalized concircular to an Einstein manifold and that its scalar curvature is prolonged covariant constant. 相似文献
17.
YAN RongMu 《中国科学 数学(英文版)》2012,(4):727-734
The purpose of the present paper is to investigate affinely equivalent Khler-Finsler metrics on a complex manifold.We give two facts (1) Projectively equivalent Khler-Finsler metrics must be affinely equivalent;(2) a Khler-Finsler metric is a Khler-Berwald metric if and only if it is affinely equivalent to a Khler metric.Furthermore,we give a formula to describe the affine equivalence of two weakly Khler-Finsler metrics. 相似文献
18.
Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests. 相似文献
19.
In this paper, we consider a class of bounded Reinhardt domains Dα(m, n1,…,nm). The Bergman kernel function K(z,z), the Bergman metric matrix T(z,z), the Cauchy-Szego kernel function S(z,ζ) are obtained. Then we prove that the formal Poisson kernel function is not a Poisson kernel function. At last, we prove that Dαis a quasiconvex domain and Dαis a stronger quasiconvex domain if and only if Dαis a hypersphere. 相似文献
20.
Let R be a prime ring of characteristic different from two with the second involution * and a an automorphism of R.An additive mapping F of R is called a generalized(α,α)-derivation on R if there exists an(α,α)-derivation d of R such that F(xy)=F(x)α(y)-α(x)d(y) holds for all x,y∈R.The paper deals with the study of some commutativity criteria for prime rings with involution.Precisely,we describe the structure of R admitting a generalized(α,α)-derivation F satisfying any one of the following prop... 相似文献