A Differentiable Sphere Theorem for Submanifolds in Euclidean Space

球面定理是整体微分几何领域的重要问题。设M为欧氏空间中的紧致子流形，如果M的里奇曲率与平均曲率满足特定的拼挤条件，本文推导出子流形M的曲率张量满足里奇流收敛定理的条件，于是M的度量在正规化里奇流下会收敛到一个常曲率度量。由此可得M微分同胚于球面空间形式。如果M还是单连通的，则M微分同胚于标准球面。Sphere theorems are important problems in global Differential geometry. Let M be a compact submanifold in Euclidean space. If the Ricci curvature and mean curvature of M satisfy a certain pinching condition, it can be deduced that the curvature tensor of submanifold M satisfies the condition of the convergence theorem of Ricci flow. Thus, the metric of M will converge to a metric of constant curvature. Hence M is diffeomorphic to a spherical space form. In addition, if M is simply connected, then M is diffeomorphic to the standard sphere.