Cheng–Shen conjecture in Finsler geometry

2020 ◽  
Vol 31 (04) ◽  
pp. 2050030
Author(s):  
M. Atashafrouz ◽  
B. Najafi
Keyword(s):  
Lie Groups ◽  
Finsler Geometry ◽  
Three Dimensional ◽  
Closed Manifold ◽  
Classification Theorem ◽  
Randers Metric ◽  
Counter Example ◽  
Randers Metrics

The well-known Cheng–Shen conjecture says that every [Formula: see text]-quadratic Randers metric on a closed manifold is a Berwald metric. The class of [Formula: see text]-quadratic Randers metrics contains the class of generalized Douglas–Weyl Randers metrics. In this paper, we give a classification of left-invariant Randers metrics of generalized Douglas–Weyl type on three-dimensional Lie groups. Based on our classification theorem, we find a counter-example for the Cheng–Shen conjecture.

scholarly journals ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

2011 ◽  
Vol 08 (03) ◽  
pp. 501-510 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Three Dimensional ◽  
Flag Curvature ◽  
Nilpotent Lie Groups ◽  
Randers Metric ◽  
Levi Civita Connection ◽  
Simply Connected ◽  
Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

scholarly journals Left invariant Randers metrics of Berwald type on tangent Lie groups

2017 ◽  
Vol 15 (01) ◽  
pp. 1850015
Author(s):  
Farhad Asgari ◽  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Randers Metric ◽  
Tangent Lie Group ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected ◽  
Randers Metrics ◽  
Tangent Bundles

Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

Geodesic orbit Randers metrics on spheres

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shaoxiang Zhang ◽  
Zaili Yan
Keyword(s):  
Finsler Geometry ◽  
Riemannian Metrics ◽  
Complete Classification ◽  
Invariant Metrics ◽  
Navigation Data ◽  
Randers Metrics

AbstractStudying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.

ON COMPLEX RANDERS METRICS

2010 ◽  
Vol 21 (08) ◽  
pp. 971-986 ◽  
Author(s):  
BIN CHEN ◽  
YIBING SHEN
Keyword(s):  
Finsler Geometry ◽  
Randers Metric ◽  
Holomorphic Curvature ◽  
Randers Metrics

A characteristic for a complex Randers metric to be a complex Berwald metric is obtained. The formula of the holomorphic curvature for complex Randers metrics is given. It is shown that a complex Berwald Randers metric with isotropic holomorphic curvature must be either usually Kählerian or locally Minkowskian. The Deicke and Brickell theorems in complex Finsler geometry are also proved.

scholarly journals On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

2021 ◽  
pp. 86-90
Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Natural Generalization ◽  
Ricci Solitons ◽  
Tensor Fields ◽  
Left Invariant Riemannian Metric ◽  
Riemannian Case

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

Left Invariant Einstein–Randers Metrics on Compact Lie Groups

2012 ◽  
Vol 55 (4) ◽  
pp. 870-881 ◽  
Author(s):  
Hui Wang ◽  
Shaoqiang Deng
Keyword(s):  
Lie Groups ◽  
Complete Classification ◽  
Simple Groups ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Navigation Data ◽  
Group Of Isometries ◽  
Randers Metrics ◽  
Naturally Reductive

AbstractIn this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

scholarly journals Classification of real three-dimensional Poisson–Lie groups

2012 ◽  
Vol 45 (17) ◽  
pp. 175204 ◽  
Author(s):  
Ángel Ballesteros ◽  
Alfonso Blasco ◽  
Fabio Musso
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Poisson Lie Groups
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