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.
Inhaltsverzeichnis
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.
