The aim of this work is threefold: First it should be a monographical work on natural bundles and natural opยญ erators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle A kT* M is in fact the value of a functor, which associates a bundle over M to each manifold M and a vector bundle homomorphism over f to each local diffeomorphism f between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that exterior derivative d transforms sections of A kT* M into sections of A k+1T* M for every manifold M can be expressed by saying that d is an operator from A kT* M into A k+1T* M
CONTENT
I. Manifolds and Lie Groups -- II. Differential Forms -- III. Bundles and Connections -- IV. Jets and Natural Bundles -- V. Finite Order Theorems -- VI. Methods for Finding Natural Operators -- VII. Further Applications -- VIII. Product Preserving Functors -- IX. Bundle Functors on Manifolds -- X. Prolongation of Vector Fields and Connections -- XI. General Theory of Lie Derivatives -- XII. Gauge Natural Bundles and Operators -- References -- List of symbols -- Author index
