J ul 2 00 9 SPHERE THEOREMS IN GEOMETRY

In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton’s Ricci flow. We also obtain here new results concerning complete manifolds with pinched curvature. 1. The Topological Sphere Theorem The sphere theorem in differential geometry has a long history, dating back to a paper by H.E. Rauch in 1951. In that paper [64], Rauch posed the question of whether a compact, simply connected Riemannian manifold M whose sectional curvatures lie in the interval (1, 4] is necessarily homeomorphic to the sphere. Around 1960, M. Berger and W. Klingenberg gave an affirmative answer to this question: Theorem 1.1 (M. Berger [3]; W. Klingenberg [48]). Let M be a compact, simply connected Riemannian manifold whose sectional curvatures lie in the interval (1, 4]. Then M is homeomorphic to Sn. More generally, Berger [4] proved that a compact, simply connected Riemannian manifold whose sectional curvatures lie in the interval [1, 4] is either homeomorphic to Sn or isometric to a compact symmetric space of rank one. K. Grove and K. Shiohama proved that the upper bound on the sectional curvature can be replaced by a lower bound on the diameter: Theorem 1.2 (K. Grove, K. Shiohama [33]). LetM be a compact Riemannian manifold with sectional curvature greater than 1. If the diameter of M is greater than π/2, then M is homeomorphic to Sn. There is an interesting rigidity statement in the diameter sphere theorem. To describe this result, suppose that M is a compact Riemannian manifold with sectional curvature K ≥ 1 and diameter diam(M) ≥ π/2. A theorem of D. Gromoll and K. Grove [27] asserts that M is either homeomorphic to 1991 Mathematics Subject Classification. Primary 53C21; Secondary 53C44.