Reduction theory for systems with symmetry deals with the problem of understanding dynamics on a manifold with an action of a Lie group. In geometric mechanics, this problem can be formulated in the Lagrangian, Hamiltonian or affine connection frameworks. While the Lagrangian and Hamiltonian formulations have been well developed, the results obtained in these setups are based on variational principles and symplectic geometry. These methods cannot be used directly in the affine connection formulation unless additional structure is available. In this thesis, 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. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-07-24 01:00:05.635

This thesis is about the structure of the boundary of the universe, i.e., points where the geometric structures of spacetime cannot be continued. In partic­ular, we study the structure of the b-boundary by B. Schmidt. It has been known for some time that the b-boundary construction has several drawbacks, perhaps the most severe being that it is often not Hausdorff separated from interior points in spacetime. In other words, the topology makes it impossible to distinguish which points in spacetime are near the singularity and which points are ‘far’ from it. The non-Hausdorffness of the b-completion is closely related to the concept of fibre degeneracy of the fibre in the frame bundle over a b-boundary point, the fibre being smaller than the whole structure group in a specific sense. Fibre degeneracy is to be expected for many realistic spacetimes, as was proved by C. J. S. Clarke. His proofs contain some errors however, and the purpose of paper I is to reestablish the results of Clarke, under somewhat different conditions. It is found that under some conditions on the Riemann curvature tensor, the boundary fibre must be totally degenerate (i.e., a single point). The conditions are essentially that the components of the Riemann tensor and its first derivative, expressed in a parallel frame along a curve ending at the singularity, diverge sufficiently fast. We also demonstrate the applicability of the conditions by verifying them for a number of well known spacetimes. In paper II we take a different view of the b-boundary and the b-length func­tional, and study the Riemannian geometry of the frame bundle. We calculate the curvature Rof the frame bundle, which allows us to draw two conclusions. Firstly, if some component of the curvature of spacetime diverges along a horizontal curve ending at a singularity, R must tend to — oo. Secondly, if the frame bundle is extendible through a totally degenerate boundary fibre, the spacetime must be a conformally flat Einstein space asymptotically at the corresponding b-boundary point. We also obtain some basic results on the isometries and the geodesics of the frame bundle, in relation to the corresponding structures on spacetime. The first part of paper III is concerned with imprisoned curves. In Lorentzian geometry, the situation is qualitatively different from Riemannian geometry in that there may be incomplete endless curves totally or partially imprisoned in a compact subset of spacetime. It was shown by B. Schmidt that a totally imprisoned curve must have a null geodesic cluster curve. We generalise this result to partially im­prisoned incomplete endless curves. We also show that the conditions for the fibre degeneracy theorem in paper I does not apply to imprisoned curves. The second part of paper III is concerned with the properties of the b-length functional. The b-length concept is important in general relativity because the presence of endless curves with finite b-length is usually taken as the definition of a singular spacetime. It is also closely related to the b-boundary definition. We study the structure of b-neighbourhoods, i.e., the set of points reachable from a fixed point in spacetime on (horizontal) curves with b-length less than some fixed number e > 0. This can then be used to understand how the geometry of spacetime is encoded in the frame bundle geometry, and as a tool when studying the structure of the b-boundary. We also give a result linking the b-length of a general curve in the frame bundle with the b-length of the corresponding horizontal curve. / digitalisering@umu

In this work is analyzed a co-frame bundle first differential extension . Received local co-ordinates transformation law of the space, constructed this space linear connection and linear co-connection. In this work is proved basis space linear connection‘s object inducts objects linear connection and linear co-connection in space. Founded inducted connection curvature tensors.