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GENERALIZED AFFINE DEVELOPMENTS

  • Park, Joon-Sik (Department of Mathematics Pusan University of Foreign Studies)
  • Received : 2014.09.04
  • Accepted : 2015.01.12
  • Published : 2015.02.15

Abstract

The (affine) development of a smooth curve in a smooth manifold M with respect to an arbitrarily given affine connection in the bundle of affine frames over M is well known (cf. S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, Vol.1). In this paper, we get the generalized affine development of a smooth curve $x_t$ ($t{\in}[0,1]$) in M into the affine tangent space at $x_0$ (${\in}M$) with respect to a given generalized affine connection in the bundle of affine frames over M.

Keywords

References

  1. C. Ehresmann, Les connexions infinitesimales dans un espace fibre differentiable, Colloque de topologie, Bruxelles, 1950.
  2. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley-Interscience, New York, 1963.
  3. K. Nomizu, Kinematics and differential geometry-Rolling a ball with a prescribed locus of contact, Tohoku Math. J. 30 (1978), 623-637. https://doi.org/10.2748/tmj/1178229921