Finsler Metrics with K = 0 and S = 0

By Zhongmin Shen. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2001

172 citations recorded by the source corpus.

Abstract In 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.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/9743e01f510c3415d01f2127c4428b62a32e2028

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In Astronomy and Astrophysics

Citations in the corpus

A class of Finsler metrics with isotropic S-curvature (2009, background): “[8][ 12 ][15]). It is known that, for a Finsler metric F of scalar flag curvature, if the S-curvature is almost isotropic, i.e.,”
Projectively flat Randers metrics with constant flag curvature (2003, background): “From examples in [BaSh] [ Sh3 ] [Sh4] [BaRo], we do not see any hope to find an explicit formula for each isometry class.”
RANDERS METRICS OF SCALAR FLAG CURVATURE (2009, methodology): “Using the navigation idea [12], one can express any Randers metric F = α + β in terms of a Riemannian metric h = √ hijyy and a vector field W = W i ∂ ∂xi by”
Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry (2008, methodology): “For example much effort has been expended on understanding constant flag curvature for Randers-Finsler metrics [6].”
On the Flag Curvature of Finsler Metrics of Scalar Curvature (2003, methodology): “We also see that F has negative constant flag curvature and bounded Cartan torsion, hence it is Riemannian according to Akbar-Zadeh’s theorem [1, 26, 29].”
On Randers metrics with isotropic S-curvature (2008, background): “Early in 2001, the first author discovered that some Randers metrics of constant flag curvature have constant S-curvature [5–6].”
Two-dimensional Finsler metrics with constant flag curvature (2002, methodology): “Moreover, they use the technique in [ Sh2 ] to construct two-dimensional Randers metrics with the Gauss curvature K = K(x) independent of the directions, a three-dimensional Randers metric on S 3 with K = 1 and a three-dimensional Randers metric on B 3 with K =− 1. We should point out that their exa”
Zermelo navigation on Riemannian manifolds (2003, background): “Shen [30] showed that the same phenomenon holds for arbitrary Riemannian backgrounds in all dimensions.”
GEODESICS IN RANDERS SPACES OF CONSTANT CURVATURE (2005, background): “ metric is strongly convex. That is, the Hessian gij(x,y) := (1 2F 2) yiyj is positive deﬁnite for all nonzero y. See [BCS00, BR04] for this result and a through treatment of Randers metrics. Z. Shen [S02] has identiﬁed Randers metrics with solutions to a navigation problem. In 1931, Zermelo posed a”
Finsler manifolds with nonpositive flag curvature and constant S-curvature (2003, background): “F is an incomplete Finsler metric on Ω with K = 0 and S = 0, but J 6= 0 [20].”
Projectively flat Finsler metrics of constant flag curvature (2002, background): “Later on, the author constructs infinitely many nonprojectively flat Finsler metrics with constant curvature [ Sh2 ][Sh3].”