On a Class of Landsberg Metrics in Finsler Geometry

2009 ◽  
Vol 61 (6) ◽  
pp. 1357-1374 ◽  
Author(s):  
Zhongmin Shen
Keyword(s):  
Open Problem ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Scalar Function ◽  
Parameter Family ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Landsberg Metric ◽  
Two Parameter

Abstract In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We show that a regular Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, ɸ = ɸ(s), for which there are a Riemannian metric 𝜶 and a 1-form ᵦ on a manifold M such that the scalar function F = 𝜶ɸ(ᵦ/𝜶) on TM is an almost regular Landsberg metric, but not a Berwald metric.

On general (α,β)-metrics of Landsberg type

2016 ◽  
Vol 13 (06) ◽  
pp. 1650085 ◽  
Author(s):  
M. Zohrehvand ◽  
H. Maleki
Keyword(s):  
General Formula ◽  
Open Problem ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Finsler Metrics

In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric [Formula: see text] and a one-form [Formula: see text]. They are called general [Formula: see text]-metrics. We have proven that, every Landsberg general [Formula: see text]-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general [Formula: see text]-metrics.

scholarly journals A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

2012 ◽  
Vol 54 (3) ◽  
pp. 637-645 ◽  
Author(s):  
XIAOHUAN MO ◽  
ZHONGMIN SHEN ◽  
HUAIFU LIU
Keyword(s):  
Scalar Curvature ◽  
Simple Proof ◽  
Finsler Geometry ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemannian Curvature

AbstractIn this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

scholarly journals On the ℒ-duality of a Finsler space with exponential metric αeβ/α

2018 ◽  
Vol 10 (1) ◽  
pp. 167-177
Author(s):  
Ramdayal Singh Kushwaha ◽  
Gauree Shanker
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Finsler Metrics ◽  
Exponential Metric

Abstract The (α, β)-metrics are the most studied Finsler metrics in Finsler geometry with Randers, Kropina and Matsumoto metrics being the most explored metrics in modern Finsler geometry. The ℒ-dual of Randers, Kropina and Matsumoto space have been introduced in [3, 4, 5], also in recent the ℒ-dual of a Finsler space with special (α, β)-metric and generalized Matsumoto spaces have been introduced in [16, 17]. In this paper, we find the ℒ-dual of a Finsler space with an exponential metric αeβ/α, where α is Riemannian metric and β is a non-zero one form.

scholarly journals On Negatively Curved Finsler Manifolds of Scalar Curvature

2005 ◽  
Vol 48 (1) ◽  
pp. 112-120 ◽  
Author(s):  
Xiaohuan Mo ◽  
Zhongmin Shen
Keyword(s):  
Scalar Curvature ◽  
Tangent Bundle ◽  
Compact Manifold ◽  
Scalar Function ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Finsler Manifolds ◽  
Global Rigidity ◽  
Negatively Curved ◽  
Projectively Flat

AbstractIn this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n ≥ 3. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

ON A CLASS OF PROJECTIVELY FLAT FINSLER METRICS OF NEGATIVE CONSTANT FLAG CURVATURE

2012 ◽  
Vol 23 (08) ◽  
pp. 1250084 ◽  
Author(s):  
XIAOHUAN MO ◽  
HONGMEI ZHU
Keyword(s):  
Structure Theorem ◽  
Finsler Geometry ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Constant Flag Curvature ◽  
Projectively Flat ◽  
Slight Generalization ◽  
Negative Constant ◽  
World Scientific Publishing

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shen's construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

On a Class of Locally Dually Flat (α,β)–Metrics

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
A. Tayebi ◽  
H. Sadeghi ◽  
E. Peyghan
Keyword(s):  
Riemannian Metric ◽  
Finsler Metric ◽  
Finsler Metrics

AbstractIn this paper, we consider locally dually flat Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We find some conditions under which such a Finsler metric is Berwaldian, Riemannian or locally Minkowskian.

Projective Ricci flat spherically symmetric Finsler metrics

2018 ◽  
Vol 29 (11) ◽  
pp. 1850078 ◽  
Author(s):  
Hongmei Zhu ◽  
Haixia Zhang
Keyword(s):  
Ricci Curvature ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Spherically Symmetric ◽  
Projective Invariant ◽  
Ricci Flat ◽  
Douglas Metric

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we characterize projective Ricci flat spherically symmetric Finsler metrics. Under a certain condition, we prove that a projective Ricci flat spherically symmetric Finsler metric must be a Douglas metric. Moreover, we construct a class of new nontrivial examples on projective Ricci flat Finsler metrics.

On generalized 4-th root metrics of isotropic scalar curvature

2018 ◽  
Vol 68 (4) ◽  
pp. 907-928 ◽  
Author(s):  
Akbar Tayebi
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Riemannian Curvature ◽  
Necessary And Sufficient Condition ◽  
Conformal Change ◽  
Riemannian Curvature Tensor ◽  
Necessary And Sufficient

AbstractBy an interesting physical perspective and a suitable contraction of the Riemannian curvature tensor in Finsler geometry, Akbar-Zadeh introduced the notion of scalar curvature for the Finsler metrics. A Finsler metric is called of isotropic scalar curvature if the scalar curvature depends on the position only. In this paper, we study the class of generalized 4-th root metrics. These metrics generalize 4-th root metrics which are used in Biology as ecological metrics. We find the necessary and sufficient condition under which a generalized 4-th root metric is of isotropic scalar curvature. Then, we find the necessary and sufficient condition under which the conformal change of a generalized 4-th root metric is of isotropic scalar curvature. Finally, we characterize the Bryant metrics of isotropic scalar curvature.

On general (α, β)-metrics with vanishing Douglas curvature

2015 ◽  
Vol 26 (09) ◽  
pp. 1550076 ◽  
Author(s):  
Hongmei Zhu
Keyword(s):  
Riemannian Metric ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Douglas Metric ◽  
Douglas Curvature

In this paper, 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 find an equation which is necessary and sufficient condition for such Finsler metric to be a Douglas metric. By solving this equation, we obtain all of general (α, β)-metrics with vanishing Douglas curvature under certain condition. Many new non-trivial examples are explicitly constructed.

Homogeneous Einstein Manifolds with Vanishing Curvature

2019 ◽  
Vol 62 (3) ◽  
pp. 525-537
Author(s):  
Libing Huang ◽  
Zhongmin Shen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Homogeneous Spaces ◽  
Parameter Family ◽  
Finsler Metrics ◽  
Einstein Manifolds ◽  
Constant Condition ◽  
Regular Solutions ◽  
Two Parameter

AbstractInfinitely many new Einstein Finsler metrics are constructed on several homogeneous spaces. By imposing certain conditions on the homogeneous spaces, it is shown that the Ricci constant condition becomes an ordinary differential equation. The regular solutions of this equation lead to a two parameter family of Einstein Finsler metrics with vanishing $S$ curvature.

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