GEODESICS IN RANDERS SPACES OF CONSTANT CURVATURE

By C. Robles. 2005

78 citations recorded by the source corpus.

Geodesies in Randers spaces of constant curvature are classified. Randers metrics have received much attention lately as solutions to Zermelo's problem of navigation; largely because this navigation structure provides the frame work for a complete classification of constant flag curvature Randers spaces. (Flag curvature is the Finslerian analog of Riemannian sectional curvature. See (BR04).) Briefly, a Randers metric is of constant flag curvature if and only if it solves Zer melo's problem of navigation on a Riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety W. See Subsection 1.1 for a sketch of the navigation problem, and Theorem 3 for an explicit statement of the classification result. The aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature. Intuitively, these paths minimize travel time across a Riemannian landscape under windy conditions. Presently we will show that these curves are given by composing geodesies of the Riemannian metric with the flow generated by W. This claim is formalized by Theorem 2. Geodesies on surfaces of constant, nonpositive curvature are illustrated in Section 3. We then turn, in Section 4, to the constant flag curvature K = 1 Randers metrics on Sn. The case of the sphere is especially interesting; it is possible to endow this closed manifold with a metric whose geodesies display distinctly non-Riemannian behaviors. For example: (1) A metric is projectively flat if every point admits coordinates in which the geodesies are straight lines. Belt r ami's theorem assures us that a Riemannian metric is of constant sectional curvature if and only if it is projectively flat. In contrast few Randers spaces of constant flag curvature are projectively flat. There are infinitely many nonisometric Randers metrics of constant, positive

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/eb18a1d1c933a78be3729b9b0b26d2fb8afe1123

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In Astronomy and Astrophysics

Citations in the corpus

Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry (2008, methodology): “For the geodesics of (1) to solve the corresponding Zermelo problem, the Randers data {aij , bi} are determined in terms of the Zermelo data {hij ,W i} as [12]: aij = λhij +WiWj λ2 , λ = 1 − hijW W j , (2) bi = − Wi λ , Wi = hijW j .”
Kropina metrics and Zermelo navigation on Riemannian manifolds (2012, background): “Moreover, in 2004, David Bao, Colleen Robles and Zhongmin Shen have obtained the necessary and sufficient conditions for a Randers space to be of constant flag curvature and completed the classification of strongly convex Randers metrics of constant flag curvature ([BRS]).”
Finsler spaces whose geodesics are orbits (2014, background): “Note that this result generalizes a theorem which was first proved in the case of Randers metrics with constant S-curvature [34].”
A Finsler geodesic spray paradigm for wildfire spread modelling (2016, methodology): “We refer to [48, 49, 50, 47, 51, 4, 52, 53, 54, 8, 55, 56, 57, 5, 58, 59, 60] for details on Randers spaces and on the Zermelo navigation problem.”
RANDERS METRICS OF SCALAR FLAG CURVATURE (2009, background): “[11] Randers metrics of scalar flag curvature 369”