Differential Geometry of Frame Bundles
It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
- Hardback | 234 pages
- 160 x 240 x 19.81mm | 528g
- 01 Jan 1989
- Dordrecht, Netherlands
- 1989 ed.
- X, 234 p.
Table of contents
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.
All in all, Differential Geometry of Frame Bundles is an excellent and modern work, offering valuable information for many readers who are interested in modern geometry and its applications.Acta Applicandae Mathematicae