Introduction: how a vector field maps a manifold into itself

In the previous sections we have developed certain aspects of index notation. This notation is often essential for dealing with actual numerical computations; but it is just as often a hindrance in developing a sound geometrical idea of what the mathematics means. We begin by defining vectors and tensors in a manner independent of any basis, and we now continue in this spirit to develop what is one of the most useful analytic tools in geometry: the lie derivative along the congruence defined by a vector field.

We have mentioned the idea of a ‘congruence’ in §2.12: a set of curves that fill the manifold, or some part of it, without intersecting. Each point in the region of the manifold M is on one and only one curve. Since each curve is a one-dimensional set of points, the set of curves is (n — l)-dimensional. (With some suitable parameterization, the set of curves is itself a manifold.) The key point from which everything else follows is that the congruence provides a natural mapping of the manifold into itself. If the parameter on the curves is λ, then any sufficiently small number Δλ defines a mapping in which each point is mapped into the one, a parameter distance Δλ further along the same curve of the congruence (see figure 3.1).