Curvature, Sphere Theorems, and the Ricci flow

Abstract

Abstract. In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature 1 is isometric to the standard sphere. Motivated by this result, Hopf posed the question whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere.In the first part of this paper, we provide a background discussion, aimed at nonexperts, of Hopf's pinching problem and the Sphere Theorem. In the second part, we sketch the proof of the Differentiable Sphere Theorem, and discuss various related results. These results employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton's Ricci flow. Hopf's pinching problem and the Sphere Theorem:A background discussion for nonexperts Differential geometry is concerned with the study of smooth n-manifolds. These are topological spaces which are locally homeomorphic to open subsets of R n by local coordinate maps such that all change of coordinate maps are smooth diffeomorphisms. The simplest examples of smooth manifolds are two-dimensional surfaces embedded in R 3 : these are subsets of R 3 with the property that a neighborhood of each point can be expressed as the graph of a smooth function over some twodimensional plane. In addition to their topological structure, such surfaces inherit two metric structures from R 3 . The most elementary metric structure is the one obtained by restricting the distance function from R 3 to M . This distance function is called the chord (or extrinsic) distance, and it depends on the embedding of M in R 3 . For our purposes, we are interested in the geodesic distance which is obtained by minimizing arc length among all curves on M joining two given endpoints. For example, when we travel along the surface of the earth from one point to another in the most efficient way, the distance we travel is the geodesic distance. The geodesic distance is an intrinsic quantity: in order to compute it, one does not need to leave the surface, and so it does not depend on the embedding of the surface into ambient space. We usually encode the geodesic distance by the first fundamental form (or metric tensor) of M . This is simply the restriction of the Euclidean inner product to each tangent plane of M . From this inner product, we can compute the lengths of tangent vectors, hence the lengths of smooth curves lying on M and, by mimimizing this length, we can find the geodesic distance function.

The Weyl tensor of gradient Ricci solitons

“…The factor of 2 is due to our convention of calculating norm. Some special cases of dimension four also appeared in [9,Proposition 4].…”

Evolution equations in Riemannian geometry

“…The Sphere Theorem has a long history, dating back to a question posed by H. Hopf in the 1940s (see [19] for a survey). The classical Sphere Theorem of Berger [4] and Klingenberg [38] asserts that any compact, simply connected Riemann manifold whose sectional curvatures all lie in the interval (1,4] is homeomorphic to the sphere.…”

Four-Manifolds With Positive Curvature

“…Such an improvement became known as the Topological Sphere Theorem. After almost 50 years, Brendle and Schoen [8] showed by an outstanding method that under the same curvature pinching of Berger and Klingenberg such a manifold must be diffeomorphic to the sphere, this result has been known as the Differentiable Sphere Theorem; see also [7]. Moreover, by combining the results of Berger [2] and Petersen and Tao [24], it is known that given a Riemannian manifold M n , there is a real number ε (unknown) such that if M n is ( 1 4 − ε)-pinched, then M n is either homeomorphic to S n or diffeomorphic to a spherical space form of rank 1.…”