Abstract

The paper is devoted to a unified approach to the theory of structured bundles .

Our idea of structuring category represents a general approach to the concept of a differential geometric category. In particular, it covers the classical examples of algebraic categories, such as those of vector spaces, affine spaces, Lie groups, principal spaces and so on. The problems of reconstruction and smoothness are studied in detail. Special attention is paid to the structured bundles of finite type.

The unifying approach to structured bundles is achieved by considering bundles, whose fibres are smoothly structured in a structuring category. The problems of reconstruction from a cocycle, from an atlas or from distinguished fibred morphisms and sections are analysed in detail.

The tangent prolongation of structured manifolds and the tangent, vertical and jet prolongation of structured bundles are studied as well.

Introduction

The theory of bundles and, in particular, vector, principal and associated bundles, is well established in current literature. However, certain aspeots are studied only in some specific frameworks; moreover, in different cases, analogous problems are approached in different ways. For instance, the reconstruction problems are often studied for vector bundles and not for affine bundles; moreover, the definitions of vector and principal bundles are usually given in two different ways.

The aim of this paper is to provide a unified approach to structured bundles. In this scheme we recover and compare the classical approaches as well.

The starting point is the concept of "structuring category". A structuring category is a faithful functor from a category to the category of smooth manifolds; thus, the objects of a structuring category are smooth manifolds along with some further information. Typical examples of structuring categories are constituted by algebraic categories, whose objects are manifolds with algebraic operations. For instance, vector spaces, Lie groups, algebras, etc. constitute algebraic categories. We remark that it is also possible to introduce the definition of the algebraic category of principal spaces, in a way which will fit our general scheme in the context of principal bundles. A more complicated definition of structuring category arises naturally for the case when an algebraic strueture is supported by several manifolds. This is the case, for instance, of affine spaces, which involve the spaces of points and vectors. An interesting problem arises with respect to important aspects of smoothness of a structuring category.

The second step is the study of unstructured bundles. Particular attention is devoted to the problem of reconstruction from a given cocycle. A preliminary possible approach to the concept of structuring on a given bundle could be achieved by the selection of a subatlas of the maximal atlas constituted by all the local trivializations. This leads us to the notion of G-bundle. However, it seems to us that this approach has conceptual limits.

We define a structured bundle to be a bundle along with a smooth structuring of its fibres into a given structuring category. This smoothness is insured by the existence of a bundle trivializing atlas compatible with the structuring category. This unified approach is suitable for all classical algebraic bundles, including Lie group bundles, vector bundles, principal bundles, affine bundles, etc. The problem of reconstruction of the algebraic structure on the bundle from the atlas, the cocycle and the distinguished fibred morphisms and sections are studied in detail. An interesting case is constituted by bundles of finite type, whose groups of automorphisms of the fibres are Lie groups. Here we use the recent theory of smooth spaces due to Froelicher and a hard analytical result by Boman. In this context, the concept of associated bundle arises naturally and the reconstruction from the cocycle is re-interpreted in terms of associated bundles.

The last chapter of this seminar text is devoted to the basic prolongation procedures in differential geometry. We explain a general scheme, which justifies that from such a point of view the algebraic structuring categories behave well as a rule. The tangent prolongation of structured manifolds and the tangent, vertical and jet prolongation of structured bundles are studied in detail.