In this chapter, we introduce one of the most powerful theorems in Riemannian geometry: H. E. Rauch's comparison theorem. It allows for direct comparison of the growth of Jacobi fields in a given Riemannian manifold M with those in a simply connected space form of constant sectional curvature in both cases, where the constant sectional curvature is an upper or lower bound of the sectional curvatures along the geodesic under consideration in M. The case where the curvature is bounded from above is quite elementary; and for the curvature bounded from below, we have already dealt with the weaker conjugate point (Bonnet–Myers) and volume (Bishop) comparison theorems (in those cases lower bounds on the Ricci curvature sufficed). So, now we turn to the strongest version, the one discussing the Jacobi fields themselves. (See the preliminary discussion in Notes II.10–II.11.)

The major applications we consider here are (i) the Heintze–Karcher volume comparison theorem for the volume of tubular neighborhoods of submanifolds of arbitrary codimension, (ii) the Alexandrov–Toponogov triangle comparison theorems, and (iii) Cheeger's finiteness theorem. Our applications are only a sample. One has, at least, a whole panoply of “sphere theorems” (see §IX.9), which were the initial major application of the Jacobi field comparison theorems – the original program of Rauch. And, most recently, one has M. Gromov's convergence theorems for Riemannian manifolds (see §IX.9).

We first list some small, but necessary, preliminaries.

Preliminaries

We fix our perennial Riemannian manifold M.