The Norwegian mathematician Sophus Lie (1842–1899) is rightly credited with the creation of one of the most fertile paradigms in mathematical physics. Some of the material discussed in the previous chapter, in particular the relation between brackets of vector fields and commutativity of flows, is directly traceable to Lie's doctoral dissertation. Twentieth-century Physics owes a great deal to Lie's ideas, and so does Differential Geometry.

Introduction

We will revisit some of the ideas introduced in Example 4.6 from a more general point of view. Just as the velocity field of a fluid presupposes an underlying flow of matter, so can any vector field be regarded as the velocity field of the steady motion of a fluid, thereby leading to the mathematical notion of the flow of a vector field. Moreover, were one to attach a marker to each of two neighbouring fluid particles, representing the tail and the tip of a vector, as time goes on the flow would carry them along, thus yielding a rate of change of the vector they define. The rigorous mathematical counterpart of this idea is the Lie derivative.

Let V : M → TM be a (smooth) vector field. A (parametrized) curve γ : I → M is called an integral curve of the vector field if its tangent at each point coincides with the vector field at that point.