Projective Changes between Generalized (&lt;i&gt;α&lt;/i&gt;, &lt;i&gt;β&lt;/i&gt;)-Metric and Randers Metric

AbstractIn the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

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Randers manifolds of positive constant curvature

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203111301 ◽

2003 ◽

Vol 2003 (18) ◽

pp. 1155-1165 ◽

Cited By ~ 3

Author(s):

Aurel Bejancu ◽

Hani Reda Farran

Keyword(s):

Riemannian Manifold ◽

Constant Curvature ◽

Positive Constant ◽

Complete Riemannian Manifold ◽

Flag Curvature ◽

Dimensional Sphere ◽

Randers Metric ◽

Constant Flag Curvature ◽

Simply Connected

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.

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ON SPHERICALLY SYMMETRIC FINSLER METRICS WITH ISOTROPIC BERWALD CURVATURE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887813500540 ◽

2013 ◽

Vol 10 (10) ◽

pp. 1350054 ◽

Cited By ~ 8

Author(s):

ENLI GUO ◽

HUAIFU LIU ◽

XIAOHUAN MO

Keyword(s):

Orthogonal Group ◽

Finsler Metric ◽

Finsler Metrics ◽

Spherically Symmetric ◽

Randers Metric

A Finsler metric F is said to be spherically symmetric if the orthogonal group O(n) acts as isometries of F. In this paper, we show that every spherically symmetric Finsler metric of isotropic Berwald curvature is a Randers metric. We also construct explicitly a lot of new isotropic Berwald spherically symmetric Finsler metrics.

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The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

Mathematics ◽

10.3390/math8112047 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2047

Author(s):

Rattanasak Hama ◽

Sorin V. Sabau

Keyword(s):

Arbitrary Direction ◽

Surfaces Of Revolution ◽

Killing Field ◽

Navigation Problem ◽

Rotational Surface ◽

Randers Metric ◽

Global Behaviour

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R×S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

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On the flag curvature of a class of Randers metric generated from the navigation problem

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2012.07.035 ◽

2013 ◽

Vol 397 (1) ◽

pp. 415-427 ◽

Cited By ~ 6

Author(s):

Qiaoling Xia

Keyword(s):

Flag Curvature ◽

Navigation Problem ◽

Randers Metric

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005257 ◽

2011 ◽

Vol 08 (03) ◽

pp. 501-510 ◽

Cited By ~ 4

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.

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