Geodesic Reduction via Frame Bundle Geometry

TL;DR

This paper explores the reduction of geodesic equations on manifolds with affine connections using frame bundle geometry, providing an intrinsic and structured approach to understanding invariant dynamics under Lie group actions.

Contribution

It introduces a novel intrinsic description of geodesic spray reduction via frame bundle geometry, linking it to tangent and vertical lifts of the symmetric product.

Findings

Derived fundamental relationship between geodesic spray, tangent lift, and vertical lift.

Provided an intrinsic geometric framework for reduction of affine connection dynamics.

Enhanced understanding of invariant geodesic structures on manifolds with Lie group actions.

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.