We survey results about the injectivity radius and sphere theorems, from the early versions of the topological sphere theorem to the authors’ most recent pinching below- theorems, explaining at each stage the new ideas involved.

Introduction

Injectivity radius estimates and sphere theorems have always been a central theme in global differential geometry. Many tools and concepts that are now fundamental for comparison geometry have been developed in this context. This survey of results of this type reaches from the early versions of the topological sphere theorem to the most recent pinching below-1 theorems. Our main concern is to explain the new ideas that enter at each stage. We do not cover the differentiate sphere theorem and sphere theorems based on Ricci curvature. In Sections 1-3 we give an account of the entire development from the first sphere theorem of H. E. Rauch to M. Berger's rigidity theorem and his pinching below-1 theorem. Many of the main results depend on subtle injectivity radius estimates for compact, simply connected manifolds.

In Section 4 we present our recent injectivity radius estimate for odd-dimensional manifolds Mn with a pinching constant below that is independent of n[Abresch and Meyer 1994]. With this estimate the restriction to even-dimensional manifolds can be removed from the hypotheses of Berger's pinching below- \ theorem.

Additional work is required in order to get a sphere theorem for odd-dimensional manifolds Mn with a pinching constant independent of n. This result and the basic steps involved in its proof are presented in Sections 5-7; details can be found in [Abresch and Meyer a].