Comparison Theorems in Riemannian Geometry

The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method. The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work: The comparison theory for the Riccati ODE, which probably goes back to L.Green [15] and which was used more systematically by Gromov [20], the triangle inequality for the Riemannian distance, the method of support function by Greene and Wu [16],[17],[34], the maximum principle of E.Hopf, generalized by E.Calabi [23], [4], and the idea of critical points of the distance function which was first used by Grove and Shiohama [21]. We have tried to present the ideas completely without being too technical. These notes are based on a course which I gave at the University of Trento in March 1994. It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures. We also thank Evi Samiou and Robert Bock for many valuable corrections.