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Ricci and flag curvatures in Finsler geometry
 In A Sampler of Riemann–Finsler Geometry, Mathematical Sciences Research Institute Publications
, 2004
"... 1.1. Finsler metrics 199 1.2. Flag curvature 207 ..."
Finsler metrics of constant positive curvature on the Lie group
 S 3 , J. Lond. Math. Soc
"... Abstract. Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group S 3. Using the Yasuda–Shimada theorem as an inspiration, we determine for each K> 1 a privileged right invariant Killing field of constant lengt ..."
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Cited by 17 (3 self)
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Abstract. Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group S 3. Using the Yasuda–Shimada theorem as an inspiration, we determine for each K> 1 a privileged right invariant Killing field of constant length. Each such Riemannian metric pairs with the corresponding Killing field to produce a yglobal and explicit Randers metric on S 3. Using the machinery of spray curvature and Berwald’s formula for it, we prove directly that the said Randers metric has constant positive flag curvature K, as predicted by the Yasuda–Shimada theorem. We also explain why this family of Finslerian space forms is not projectively flat. 1.
On the flag curvature of Finsler metrics of scalar curvature
 J. London Math. Soc
, 2003
"... The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several nonRiemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the Scurvature, whi ..."
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Cited by 17 (10 self)
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The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several nonRiemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the Scurvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain nonRiemannian quantities are isotropic. Using the obtained formula for the flag curvature, we classify locally projectively flat Randers metrics with isotropic Scurvature. 1
Geodesics in Randers spaces of constant curvature
, 2005
"... Abstract. Geodesics in Randers spaces of constant curvature are classified. 1. ..."
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Abstract. Geodesics in Randers spaces of constant curvature are classified. 1.
The Euler–Lagrange PDE and Finsler metrizability
 Houston J. Math
"... Abstract. In this paper we investigate the following question: under what conditions can a secondorder homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the EulerLagrange partial differential system on the energy function can be reduced to ..."
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Abstract. In this paper we investigate the following question: under what conditions can a secondorder homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the EulerLagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vectorfields. We also consider the Landsberg metrizability problem and prove similar results. This reduction is a significant step in solving the problem whether or not there exists a nonBerwald Landsberg space. 1.
Dynamically convex Finsler metrics and Jholomorphic embeddings of asymptotic cylinders
, 2007
"... We explore the relationship between contact forms on S³ defined by Finsler metrics on S² and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in [9, 10]. We show that a Finsler metric on S² with curvature K ≥ 1 and with all geodesic loops of length> π is dynamically convex and hence it ..."
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Cited by 9 (1 self)
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We explore the relationship between contact forms on S³ defined by Finsler metrics on S² and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in [9, 10]. We show that a Finsler metric on S² with curvature K ≥ 1 and with all geodesic loops of length> π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct Jholomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on S² with K = 1 thus complementing the results obtained in [8].
Finsler Conformal LichnerowiczObata conjecture
, 802
"... We prove the Finsler analog of the conformal LichnerowiczObata conjecture showing that a complete and essential conformal vector field on a nonRiemannian Finsler manifold is a homothetic vector field of a Minkowski metric. MSC 2000: 53A30; 53C60 Key words: essential conformal vector field, conform ..."
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Cited by 9 (3 self)
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We prove the Finsler analog of the conformal LichnerowiczObata conjecture showing that a complete and essential conformal vector field on a nonRiemannian Finsler manifold is a homothetic vector field of a Minkowski metric. MSC 2000: 53A30; 53C60 Key words: essential conformal vector field, conformal transformations, Finsler metrics, LichnerowiczObata conjecture 1 Definitions and results In this paper a Finsler metric on a smooth manifold M is a function F: TM → R≥0 satisfying the following properties: 1. It is smooth on TM \ TM0, where TM0 denotes the zero section of TM, 2. For every x ∈ M, the restriction FTxM is a norm on TxM, i.e., for every ξ, η ∈ TxM and for every nonnegative λ ∈ R≥0 we have (a) F(λ · ξ) = λ · F(ξ), (b) F(ξ + η) ≤ F(ξ) + F(η),
Curvature and global rigidity in Finsler manifolds
 Houston J. Math
"... Abstract. We present some strong global rigidity results for reversible Finsler manifolds. Following E ́ Cartan’s definition (1926), a locally symmetric Finsler metric is one whose curvature is parallel. These spaces strictly contain the spaces such that the geodesic reflections are local isometrie ..."
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Abstract. We present some strong global rigidity results for reversible Finsler manifolds. Following E ́ Cartan’s definition (1926), a locally symmetric Finsler metric is one whose curvature is parallel. These spaces strictly contain the spaces such that the geodesic reflections are local isometries and also constant curvature manifolds. In the case of negative curvature, we prove that the locally symmetric Finsler metrics on compact manifolds are Riemannian and this, therefore, extends A. Zadeh’s rigidity result. Our approach uses dynamical properties of the flag curvature. We also give a full generalization of the Ossermann Sarnak minoration of the metric entropy of the geodesic flow. In positive curvature, we just announce some partial results and remarks concerning Finsler metrics of curvature +1 on the 2sphere. We show that in the reversible case the geodesic flow is conjugate to the standard one. We also observe that a condition of integral geometry (of Radon type) forces such a metric to be Riemannian. This indicates a deep
Deformations and Hilbert’s Fourth Problem
"... In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1form. We classify those of projectively flat in dimension n ≥ 3 by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some ..."
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Cited by 4 (4 self)
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In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1form. We classify those of projectively flat in dimension n ≥ 3 by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric. 1