Randers Metrics and Geodesics

  • Xinyue Cheng
  • Zhongmin Shen


Let M be an n-dimensional manifold. For a point xM, let T x M denote the tangent space at x. The tangent bundle \( TM: = \bigcup\limits_{x \in M} {T_x M} \) consists of all tangent vectors on M with natural manifold structure. We denote elements in TM by (x, y), where yT x M. If (x i ) is a local coordinate system in M, then \( \left\{ {\frac{\partial } {{\partial x^i }}} \right\}\) is a local natural basis for TM. It induces a standard local coordinate system (x i , y i ) in TM by \( \left. {y = y^i \frac{\partial } {{\partial x^i }}} \right|_x \). We shall not distinguish between x and its coordinates (x i ) and (x, y) and its coordinates (x i , y i ) in the standard local coordinate system in TM.


Finsler Manifold Finsler Metrics Navigation Data Navigation Problem Minkowski Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [AnInMa]
    P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, 1993.Google Scholar
  2. [BaRoSh]
    D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geometry, 66(2004), 377–435.MathSciNetzbMATHGoogle Scholar
  3. [ChSh]
    S. S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific Publishers, 2005.Google Scholar
  4. [Ra]
    G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev., 59(1941), 195–199.MathSciNetCrossRefGoogle Scholar
  5. [Rapc]
    A. Rapcsák, Über die bahntreuen Abbildungen metrischer Räume, Publ. Math. Debrecen, 8(1961), 285–290.MathSciNetzbMATHGoogle Scholar
  6. [Ro]
    C. Robles, Geodesics in Randers spaces of constant curvature, Trans. Amer. Math. Soc., 359(4)(2007), 1633–1651.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Sh]
    Z. Shen, Finsler metrics with K = 0 and S = 0, Canad. J. Math., 55(1)(2003), 112–132.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xinyue Cheng
    • 1
  • Zhongmin Shen
    • 2
  1. 1.School of Mathematics and StatisticsChongqing University of TechnologyLijiatuo, ChongqingChina
  2. 2.Department of Mathematical SciencesIndiana University-Purdue University Indianapolis (IUPUI)IndianapolisUSA

Personalised recommendations