Differential Geometry of Frame Bundles

1 The Functor Jp1.- 1.1 The Bundle Jp1M ? M.- Functorial properties of Jp1.- Canonical lifts of vector fields to Jp1M.- Two particular cases.- Diffeomorphisms ? Mp,1 and ? M1,p.- 1.2 Jp1G for a Lie group G.- Jp1G acting on Jp1M.- 1.3 Jp1V for a vector space V.- Jp1g for a Lie algebra g.- 1.4 The embedding jp.- V = Rn.- 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.- Linear endomorphisms.- Bilinear forms.- Linear groups.- 3 Vector-valued differential forms.- 3.1 General Theory.- Particular cases.- V =? s1Rn.- V =? sRn.- 3.2 Applications.- Prolongation of functions and forms.- Complete lift of functions and tensor fields.- Prolongation of G-structures.- 4 Prolongation of linear connections.- 4.1 Forms with values in a Lie algebra.- 4.2 Prolongation of connections.- Local expressions.- Covariant differentiation operators.- 4.3 Complete lift of linear connections.- Parallelism.- 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.- G-structures from (1, 1)-tensors.- G-structures from (0, 2)-tensors.- General tensor fields.- 6 Horizontal lifts.- 6.1 General theory.- 6.2 Applications.- Tensor fields.- Linear connections.- Geodesics.- Covariant derivative.- Canonical flat connection on FM.- Derivations.- 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.- f-structures on FM.- Almost Hermitian structure.- Harmonic frame bundle maps.- 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.- Example 1: f(3, 1)-structure on FM.- Example 2: f(3, -1)-structure on FM.- Example 3: f(4,2)-structure on FM.- Example 4: f(4, -2)-structure on FM.- Example 5: A family of examples.- 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.- Local expressions.- Examples of linear connections.- Notation for sections.- 9.2 Principal bundle connections.- Summary for the principal bundle of frames.- 9.3 Systems of connections.- Examples of systems of linear connections.- 9.4 Universal Connections.- 9.5 Applications.- Universal holonomy.- Weil's Theorem.- Spacetime singularities.- Parametric models in statistical theory.- 10 The Functor Jp2.- 10.1 The Bundle Jp2M ? M.- Functorial properties of Jp2.- 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.- Algebraic preliminaries.- Diagonal lifts of 1-forms.- Diagonal lifts of (1, 1)-tensor fields.- Diagonal lifts of (0, 2)-tensor fields.- F2M for an almost Hermitian manifold M.- 10.8 Natural prolongations of G-structures.- Imbedding of Jn2FM into FF2M.- Applications.- Linear endomorphisms.- Bilinear forms.- 10.9 Diagonal prolongation of G-structures.- Applications.- Linear endomorphisms.- Bilinear forms.