Shape of manifolds with positive Ricci curvature
Shape of manifolds with positive Ricci curvature
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DOI：
10.1007/s002220050049
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发表时间：
1996-01
影响因子：
3.1
通讯作者：
T. Colding
中科院分区：
文献类型：
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作者：
T. Colding
本文的主要目的是证明，里奇曲率大于或等于 (n− 1) 的 n 维流形（在格罗莫夫-豪斯多夫拓扑中）接近单位 n 球体，其体积接近于球体的体积。这显示了[C1]中定理的逆命题。即与[C1]一起表明，当且仅当体积接近球体时，利玛窦曲率大于或等于(n−1)的n流形接近球体。特别地，通过[P]，这样的流形与球体同胚。此外，作为这一点的应用和[C1]的结果，我们证明了一个半径定理，即如果利奇曲率大于或等于(n−1)的n流形，其半径几乎等于；那么体积就接近球体的体积。为了获得这些结果，我们进一步开发和应用 [C1] 的估计。 [C1] 中主要关注的是大尺度几何，而本文主要关注的是小尺度几何。让！ n 是 n 圆球的体积，Sn；截面曲率一。
The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n− 1) which is close (in the Gromov–Hausdorff topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n− 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n− 1) has radius almost equal to; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let! n be the volume of the round n-sphere, Sn; with sectional curvature one.