Differential Geometry of Frame Bundles

1 The Functor Jp1.- 1.1 The Bundle Jp1M ? M.- 1.2 Jp1G for a Lie group G.- 1.3 Jp1V for a vector space V.- 1.4 The embedding jp.- 2 Prolongation of G-structures.- 2.1 Imbedding of Jn1FM into FFM.- 2.2 Prolongation of G-structures to FM.- 2.3 Integrability.- 2.4 Applications.- 3 Vector-valued differential forms.- 3.1 General Theory.- 3.2 Applications.- 4 Prolongation of linear connections.- 4.1 Forms with values in a Lie algebra.- 4.2 Prolongation of connections.- 4.3 Complete lift of linear connections.- 4.4 Connections adapted to G-structures.- 4.5 Geodesics of ?C.- 4.6 Complete lift of derivations.- 5 Diagonal lifts.- 5.1 Diagonal lifts.- 5.2 Applications.- 6 Horizontal lifts.- 6.1 General theory.- 6.2 Applications.- 7 Lift GD of a Riemannian G to FM.- 7.1 GD, G of type (0,2).- 7.2 Levi-Civita connection of GD.- 7.3 Curvature of GD.- 7.4 Bundle of orthonormal frames.- 7.5 Geodesics of GD.- 7.6 Applications.- 8 Constructing G-structures on FM.- 8.1 ?-associated G-structures on FM.- 8.2 Defined by (1,1)-tensor fields.- 8.3 Application to polynomial structures on FM.- 8.4 G-structures defined by (0,2)-tensor fields.- 8.5 Applications to almost complex and Hermitian structures.- 8.6 Application to spacetime structure.- 9 Systems of connections.- 9.1 Connections on a fibred manifold.- 9.2 Principal bundle connections.- 9.3 Systems of connections.- 9.4 Universal Connections.- 9.5 Applications.- 10 The Functor Jp2.- 10.1 The Bundle Jp2M ? M.- 10.2 The second order frame bundle.- 10.3 Second order connections.- 10.4 Geodesics of second order.- 10.5 G-structures on F2M.- 10.6 Vector fields on F2M.- 10.7 Diagonal lifts of tensor fields.- 10.8 Natural prolongations of G-structures.- 10.9 Diagonal prolongation of G-structures.