Riv.Mat.Univ.Parma (4) 17* (1991)


Principal bundles in action

Pages 1-65
Received: 20 July 1992
Mathematics Subject Classification: 55R10

Abstract The notion of principal bundle represents a fundamental frame in contemporary geometry. It provides both a unitary description and a deeper comprehension of a large class of phenomena, ranging from existence of further structures on a differentiable manifold (Riemann, conformal, almost-complex etc.) to the theory of connections in the modern approach to Differential Geometry. The aim of this paper is to gather and develop the basic features and results in the theory of principal bundles and connections on them: we hope in this way to contribute to fill a gap in the litterature, which, in spite of the increasing role played by principal bundels, seems to be quite reticent about general expositions on the subject and, therefore to provide a somehow useful tool. The plan of the paper is the following: chapter 1 is devoted to the general theory of principal bundles. In Section 1 we define principal bundles, we describe some fundamental examples, properties and constructions, stressing the role of the group action. In Section 2 we take care of associated fibre bundles and, in particular, of vector bundles, presenting a principal bundels approach to canonical vector bundles constructions (dual bundle, subbundles, direct sum and tensor products, morphisms). Finally, in Section 3 we consider the gauge group of a principal bundle. The subject of Chapter 2 is the thoery of connections. Section 4 provides the basic features of the teory of connections, including several standard and non Section 5 we describe pseudotensorial and tensorial forms and, by means of the results achieved in Chapter 1, we consider the induced principal bundels view point of vector bundles values forms. This includes various presentations of the exterior covariant differential operator, curvature, horizontal/vertical splitting on associated bundles and basic geometric interpretations of covariant derivative. Section 6 recalls some of the results of the holonomy theory and Section 7 is concerned with the behaviour of connection with respect to bundle morphisms and some applications (e. g. generalized Codazzi-Mainardi equation (7.1), or reduction of connections). In Section 8 we introduce a scalar product on the space of tensorial forms, we define Hodge's * operator and covariant codifferential operator, pointing out some of their fundamental properties, and we describe the basic gauge-theoretic results in the theory of characteristic classes. Finally, Section 9 enlights some of the special features of linear connections and Section 10 provides a schort account of the theory of moduli spaces of connections. Manifolds and maps between them will be understood to be C . H(M) will denote the Lie algebra of C vector fields of the manifold M. Let G be a Lie group and ℘ be its Lie algebra. Then

Ad(a): G→G     is defined by Ad(a)(g)=aga -1
Ad: G→Aut ℘     is defined by ad(a)=(Ad(a))*[e]
αδ: ℘→End ℘    is defined by αδ=(ad)*[e].