III - Riemannian Volume

Summary

We begin here our foray into the global theory – where we consider the full Riemannian manifold M. Our very first steps, in this chapter, are devoted to describing the cut locus of a point. In short, for each unit tangent vector ξ at a point p, the cut point of p along the geodesic γ_ξ emanating from p is the point along γ_ξ after which γ_ξ no longer minimizes distance from p. The collection of such cut points of p, the cut locus C(p) of p, determine the topology of M since M \ C(p) is diffeomorphic to an n–disk.

In integration theory, the major topic of the chapter, the cut locus C(p) has measure equal to 0, so the topology of M may be effectively disregarded at the early stages of study of the influence the geometry of M has on the volume measure of M. But one cannot be so cavalier. The Gauss–Bonnet theorem (see §V.1) implies that when a connected compact surface has constant Gauss curvature – 1, then knowledge of the area (2–dimensional volume) of the surface is equivalent to knowledge of the topology of the surface.

Nevertheless, our study in this chapter does not devote itself to the development of this interplay between volume and topology. Rather, it starts at a more elementary level. It continues the development of the comparison theorems of Chapter II. The basic idea is that when curvature influences the rate at which geodesics emanating from the same point separate, it automatically influences the rate at which the volume grows. Thus, the study of the geodesics is finer than the study of the volume.