In this chapter we can begin our study of the invariant calculus. The concept from which all else derives is that of a moving frame. We use the definition and construction as detailed by Fels and Olver (1998, 1999). Although the term ‘moving frame’, or ‘repère mobile’ is associated with Èlie Cartan (1953), the idea was used, albeit implicitly, long before. A pre-Cartan history of the subject is given by Akivis and Rosenfeld (1993), and the Fels and Olver papers have a more recent historical overview. The definition of a moving frame used here has the major advantage that it can be applied to both smooth and discrete problems. In particular, there is no need for any of the paraphernalia of Differential Geometry such as exterior calculus, frame bundles and connections.

Moving frames

The original problem solved by moving frames was the equivalence problem, ‘when can two surfaces be mapped one to the other, under a coordinate transformation of a particular type?’ It turns out there are many problems which can be formulated this way. One is the classification problem of differential equations. If you have a differential equation to solve and a database of solved equations, it is only sensible to ask, is there a coordinate transformation that takes my equation to one of the solved ones? Viewing differential equations as surfaces in (x, u, ux, uxx, …) space, you might then apply moving frame theory.

In this chapter we examine briefly the details of the technical definition of a Lie group. This chapter can be skipped on a first reading of this book. Eventually, however, taking a small amount of time to be familiar with the the concepts involved will pay major dividends when it comes to understanding the proofs of the key theorems.

By definition, Lie groups are locally Euclidean, so we can use tools we know and love from calculus to study functions, vector fields and so on that can be defined on them. Thus, we study differentiation on a Lie group. There are at least three important cases to consider. The first involves understanding the intrinsic definition of tangent vectors. These ideas inform every other understanding of a tangent vector, so we do that first. A second and simpler line of argument is strictly for matrix presentations, while a third treats tangent vectors as linear, first order differential operators. We will need all three.

The major theorem we prove is that the set of tangent vectors at any given point g ∈ G is in one-to-one correspondence with the set of one parameter subgroups of G. After a discussion of the exponential map in its various guises, we end the chapter with a discussion of concepts analogous to tangent vectors, one parameter subgroups and the exponential map for transformation groups.

In the previous chapter, we discussed the tangent structure on a Lie group, and the relationship between the set of one parameter subgroups and the tangent space TeG at the identity element. The most striking feature of the tangent space at the identity of a Lie group is the existence of a natural product, called a Lie bracket, so that TeG is an algebra; the Lie algebra of the Lie group.

Since Lie groups arise in different formulations, so does the appearance of the bracket in the Lie algebra. However, they all follow from the one formula for the Lie bracket of two vector fields on ℝn which we consider in the first section. The geometric formulation looks unusable in practice, so we ‘deconstruct’ it to make it easily computable, prove some of its properties and discuss the all important Frobenius Theorem. We then derive the Lie algebra bracket for a general Lie group in Section 3.2, giving details in the two main cases of interest, matrix groups in Section 3.2.1 and transformation groups in Section 3.2.2. Although many authors simply give the formulae for the Lie bracket in these two cases as the definition of the Lie bracket, and readers only needing to compute can skip straight to these formulae, it is both interesting and helpful to know that in fact they are both instances of the same geometric construction.

I first became enamoured of the Fels and Olver formulation of the moving frames theory when it helped me solve a problem I had been thinking about for several years. I set about reading their two 50-page papers, and made a 20-page handwritten glossary of definitions. I was lucky in that I was able to ask Peter Olver many questions and am eternally grateful for the answers.

I set about solving the problems that interested me, and realised there were so many of them that I could write a book. I also wanted to share my amazement at just how powerful the methods were, and at the essential simplicity of the central idea. What I have tried to achieve in this book is a discussion rich in examples, exercises and explanations that is largely accessible to a graduate student, although access to a professional mathematician will be required for some parts. I was extremely fortunate to have six students read through various drafts from the very beginning. The comments and hints they needed have been incorporated, and I have not hesitated to put in a discussion, example, exercise or hint that might be superfluous to a professional.

There is a fair amount of original material in this book. Even though some of the problems addressed here have been solved using moving frames already, I have re-proved some results to keep both solution methods and proofs within the domain of the mathematics developed here.

The curve completion problem

Consider the ‘curve completion problem’, which is a subproblem of the much more complex ‘inpainting problem’. Suppose we are given a partially obscured curve in the plane, as in Figure 0.1, and we wish to fill in the parts of the curve that are missing. If the missing bit is small, then a straight line edge can be a cost effective solution, but this does not always give an aesthetically convincing look. Considering possible solutions to the curve completion problem (Figure 0.2), we arrive at three requirements on the resulting curve:

• it should be sufficiently smooth to fool the human eye,

• if we rotate and translate the obscured curve and then fill it in, the result should be the same as filling it in and then rotating and translating,

• it should be the ‘simplest possible’ in some sense.

The first requirement means that we have boundary conditions to satisfy as well as a function space in which we are working. The second means the formulation of the problem needs to be ‘equivariant’ with respect to the standard action of the Euclidean group in the plane, as in Figure 0.3. This condition arises naturally: for example, if the image being repaired is a dirty photocopy, the result should not depend on the angle at which the original is fed into the photocopier.