Sphere theorem

Abstract: A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.

Abstract: For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility λ= max {F(−X)|F(X)=1}≥1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n≥3 with reversibility λ and with flag curvature \({{{{\left({{1-\frac{{1}}{{1+\lambda}}}}\right)}}^2 < K \le 1}}\) is homotopy equivalent to the n-sphere.

Abstract: In 1982, R Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture Furthermore, various convergence theorems have been established This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow The proofs rely mostly on maximum principle arguments Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold whose sectional curvatures all lie in the interval (1,4] is diffeomorphic to a spherical space form This question has a long history, dating back to a seminal paper by H E Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen This text originated from graduate courses given at ETH Zurich and Stanford University, and is directed at graduate students and researchers The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required

Abstract: Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S × S, also fit into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking an isometric quotient of a compact Lie group equipped with a biinvariant metric and another a gluing procedure due to Cheeger and recently significantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and find new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones. The techniques used to construct them are fairly simple. In addition to the above, the main tool is a deformation described by Cheeger that, when applied to nonnegatively curved manifolds, tends to increase curvature. Such Cheeger deformations can be considered as the unifying theme of this survey. We can thus be fairly explicit in the proof of the existence of all known examples which should make the basic material understandable at an advanced graduate student level. It is the hope of this author that it will thus encourage others to study this beautiful subject. This survey originated in the Rudolph Lipschitz lecture series the author gave at the University of Bonn in 2001 and various courses taught at the University of Pennsylvania.