Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 020, 17 pages      arXiv:1002.4156      https://doi.org/10.3842/SIGMA.2010.020

Geodesic Reduction via Frame Bundle Geometry

Ajit Bhand
Department of Mathematics, University of Oklahoma, Norman, OK, USA

Received October 12, 2009, in final form February 18, 2010; Published online February 22, 2010

Abstract
A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.

Key words: affine connection; geodesic spray; reduction; linear frame bundle.

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