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Based mainly on examples of interest in mechanics, we define the notion of a polite group action. One may view this as not only trying to give a more general notion than properness of a group action, but also to more fully understand the role of invariant functions in describing just about everything of interest in reduction.We show that a polite action of a symmetry group of a dynamical system admits reduction and reconstruction.

This paper is a rigorous study of two dual pairs of momentum maps arising in the context of fluid equations whose configuration Lie group is the group of automorphisms of a trivial principal bundle, generically called here non-abelian fluids. It is shown that the actions involved are mutually completely orthogonal, which directly implies the dual pair property.

Let E be a homogeneous vector bundle over the abelian variety A and let Aut A (E) be the (algebraic) group of automorphims of E as a vector bundle. Then the fiber over 0 is a Aut A (E)-module. We prove that E is the induced space of this action to the whole group of automorphims of the homogeneous vector bundle. The principal significance of this result is that it allows one to obtain results about the structure of E and it provides some insight into the structure of its endomorphisms monoid.

In previous chapters we have described how the notion of symmetry of a physical theory forced us to change the definition of derivative of a field so as to make the action principle invariant. The invariance of the action means that it must be the same for all observers (when considering coordinate transformations) and for the observable fields (when referring to gauge transformations). It has to be such invariant to fulfill the purposes of Maupertuis, Euler, Lagrange, and Hamilton. The modification of the derivative is quite intuitive when we are talking about coordinate transformations, but it is less intuitive when we are talking about field theory. In this chapter we will introduce the basic tools to the formal definition of gauge covariant derivatives.

We saw in Chapter 2 that the concept of homotopy plays a central role in bundle theory. Although we have so far dealt only with differentiable maps, it is more convenient, when working with homotopies, to consider continuous maps, and we will do so in this chapter. One purpose of this section is to try and convince the reader that we are not introducing new objects when, for example, we consider the pullback f*ξ of a bundle via a continuous map f: Explicitly, we will show that any continuous map between manifolds is homotopic to a differentiable one, and the latter can be chosen to be arbitrarily close to the original one.

There are many more classical applications of homotopy theory to geometry, but the central theme here being differential geometry, we wish to return to the differentiable category. Our next endeavor is to try and understand how bundles fail to be products by parallel translating vectors around closed loops. This depends of course on what is meant by “parallel translation” (which is explained in the section below), but roughly speaking, if parallel translation always results in the original vector, then the bundle is said to be flat. Otherwise, it is curved, and the amount of curvature is measured by the holonomy group, which tallies the difference between the end product in parallel translation and the original one. In this chapter, all maps and bundles are once again assumed to be differentiable.

A continuous map p: X → Y has the lifting property with respect to a pair of topological spaces (Z, A) if for every commutative diagram of the form there exists a continuous map \(\Bbbk \) : Z → X such that pk = g and ki = f.

This chapter returns to a group-theoretical context, namely, a systematic study of CS associated to group representations. After a general definition of these covariant CS, we describe the well-known Gilmore–Perelomov CS, as well as vector and matrix CS. Some generalizations are mentioned, including continuous semi-frames. We conclude the chapter by a thorough description of two interesting cases. First we treat CS on spheres constructed via heat kernels (such CS are not of the Gilmore–Perelomov type). Next we turn to CS on conformal classical domains, i.e., classical domains associated to the conformal group SO(n, 2).

The Im ℍ-valued 1-form \(\omega = \mathrm{Im}\,(\bar{{q}}^{1}{\mathit{dq}}^{1} +\bar{ {q}}^{2}{\mathit{dq}}^{2})\) will occupy center stage for much of the remainder of our story. We begin by adopting its two most important properties ((5.9.10) and (5.9.11)) as the defining conditions for a connection on a principal bundle.

It sometimes transpires that mathematics and physics, pursuing quite different agendas, find that their intellectual wanderings have converged upon the same fundamental idea and that, once it is recognized that this has occurred, each breathes new life into the other. The classic example is the symbiosis between General Relativity and Differential Geometry. As the Singularity Theorems of Hawking and Penrose (see [N2]) amply attest, the results of such an interaction can be spectacular. The story we have to tell is of another such confluence of ideas, more recent and perhaps even more profound. Our purpose in this preliminary chapter is to trace the physical and geometrical origins of the notion of a “gauge field” (known to mathematicians as the “curvature” of a “connection on a principal bundle” ). We will not be much concerned yet with rigorously defining the terms we use, nor will we bother to prove most of our assertions. Indeed, much of the remainder of the book is devoted to these very tasks. We hope only to offer something in the way of motivation.

Every representation M of a topological group G and every principal bundle α over a space X determine a fibre bundle α[M] over X that admits the structure of a vector bundle. For a given α the function that assigns α[M] to M prolongs to a group morphism R(G) → K(X), where R(G) is the representation ring of G. We study K(X), using this morphism; in particular, properties of operations in K(X) can be derived from properties of operations in R(G).

Apart from the previous chapter, the theory of fibre bundles in this book is a theory over an arbitrary space. Even the relation to manifolds in Chapter 18 is treated from a topological point of view, but in the context of smooth manifolds and vector bundles we can approach Chern classes using constructions from analysis. This idea, which goes back to a letter from A. Weil (see A. Weil Collected papers, Volume III, pages 422–36 and 571–574), involves choosing a connection or covariant derivative on the complex vector bundle, defining the curvature 2-form of the connection, and representing the characteristic class as closed 2q-form which is a polynomial in the curvature form. This proceedure is outlined in this chapter.

From now on we deal with a certain orbit space M. This “moduli space” is the set of solutions to the self-dual Yang-Mills equations, but divided out by a natural equivalence. The self-dual Yang-Mills equations are an elliptic (except along orbits of the group of gauge transformations) system of partial differential equations for a connection on a vector bundle over a smooth 4-manifold. The group of gauge transformations is the group of natural equivalences in the vector bundle, hence the natural equivalence in the problem, and so it acts on the space of solutions to the self-dual Yang-Mills equations.

It is often pertinent for the statistician to investigate how various quantities, such as combinants or arrays of combinants or confidence regions, behave under reparametrizations of the given statistical model. In particular, it is of interest to know whether the quantities are parametrization invariant, as discussed in section 1.3.