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First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over ${\mathbb {R}}^{n}$ are exactly the Minkowski spaces (Theorems 5 and 6). 相似文献
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Some constructions of projectively flat Finsler metrics 总被引:6,自引:0,他引:6
MO Xiaohuan SHEN Zhongmin & YANG Chunhong LMAM School of Mathematical Sciences Peking University Beijing China Department of Mathematical Sciences Indiana University-Purdue University Indianapolis IN - USA Department of Mathematics Inner Mongolia University Hohhot China 《中国科学A辑(英文版)》2006,49(5):703-714
In this paper, we find some solutions to a system of partial differential equations that characterize the projectively flat Finsler metrics. Further, we discover that some of these metrics actually have the zero flag curvature. 相似文献
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In this paper, we classify locally projectively flat general -metrics on an -dimensional manifold if α is of constant sectional curvature and . Furthermore, we find equations to characterize this class of metrics with constant flag curvature and determine their local structures. 相似文献
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In this paper, we hall discuss the projective flatness of complex Finsler metrics by investigating the geometry of projective bundles associated with a holomorphic vector bundle. 相似文献
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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. 相似文献
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Yongdo Lim 《Mathematische Annalen》2000,316(2):379-389
Let V be a simple Euclidean Jordan algebra with an associative inner product and let be the corresponding symmetric cone. Let be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by
is a (the automorphism group of )-invariant complete metric on and it coincides with a natural Finsler distance on We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of .
Received: 24 May 1999 / in final form: 15 March 1999 相似文献
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In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their
geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler
metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to
symmetric Finsler spaces and hence are Berwaldian. 相似文献
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In this paper, we give the equivalent PDEs for projectively flat Finsler metrics with constant flag curvature defined by a Euclidean metric and two 1-forms. Furthermore, we construct some classes of new projectively flat Finsler metrics with constant flag curvature by solving these equivalent PDEs. 相似文献
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In this paper, we consider a class of Finsler metrics which obtained by Kropina change of the class of generalized m-th root Finsler metrics. We classify projectively flat Finsler metrics in this class of metrics. Then under a condition, we show that every projectively flat Finsler metric in this class with constant flag curvature is locally Minkowskian. Finally, we find necessary and sufficient condition under which this class of metrics be locally dually flat. 相似文献
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Chang-Wan Kim 《Archiv der Mathematik》2007,88(4):378-384
We showed that any compact locally symmetric Finsler metric with positive flag curvature must be Riemannian.
Dedicated to Professor Karsten Grove on the occassion of his sixtieth birthday
Received: 8 May 2006 相似文献
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In this paper, we study locally projectively flat Finsler metrics with constant flag curvature K. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when K=0, K=−1 and K=1 are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature K=1 determined by a Minkowski norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before. 相似文献
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In this paper, we study a class of Finsler metrics defined by a Riemannian metric and 1-form. We classify those metrics which are projectively flat with weakly isotropic flag curvature. 相似文献
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《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(10):1127-1132
É. Cartan introduced in 1926 the Riemannian locally symmetric spaces, as the spaces whose curvature tensor is parallel. They also owe their name to the fact that, for each point, the geodesic reflexion is a local isometry. The aim of this Note is to announce a strong rigidity result for Finsler spaces. Namely, we show that a negatively curved locally symmetric (in the first sense above) Finsler space is isometric to a Riemann locally symmetric space. 相似文献