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Geodesic reduction via frame bundle geometry. (English) Zbl 1189.53013

Summary: 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.

MSC:

53B05 Linear and affine connections
53C05 Connections (general theory)
53C22 Geodesics in global differential geometry
58D19 Group actions and symmetry properties
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