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

Vector bundles , which basic space is smooth n-dimensional space with affine connection, are exploring in this paper. It was proved that linear connection of such bundle defines affine connection, there were found curvature objects of these connections. It was also proved that linear connection induces inside almost product’s structure in bundle , there were explored integrable conditions of this structure and defined criterions on affine connections that are associated with these structures.

The vector bundles with the basic structure space with affine connection. It is shown that in the present bundles the linear inducts the affine connection, and the curvature of the objects of the present connection is traced. Having defined the concept of the first differential extension of the vector bundles, an indication is made that the linear connection of vector bundles inducts the elongated linear connection of space and linear co-connection, expression form of the linear co-connection components and their interrelation. There are derived commutative formulas of the inducted connection and forms of its components of curvature objects.

Darbe nagrinėjamos koliestinės sluoksniuotės su apibrėžta metrika. Parodyta kaip šių erdvių metrinis tenzorius indukuoja tiesinę ir afiniąją sietį, surasti šių siečių kreivumo objektai. Įrodyta, kad šiose sluoksniuotėse egzistuoja vidinės homogeninės beveik kompleksinės struktūros. Nustatytos šių struktūrų integruojamumo sąlygos, surastos sąlygos, kad šį struktūra būtų Kelerio tipo. / In this paper cotangent bundle with determinate metrics are analysed. It is shown how metrical tensor of those spaces induces linear and affine connections, also objects of those connections‘ curvature are found. It is proved that intrinsic homogeneous almost complex structure exists in that bundle. Conditions of those structure integration are identified as well as conditions for this structure to be Kahler‘s type are found.