欧氏空间子流形的一个微分球面定理
A Differentiable Sphere Theorem for Submanifolds in Euclidean Space
DOI: 10.12677/pm.2025.155156, PDF,   
作者: 常小芳：重庆师范大学数学科学学院，重庆
关键词: 子流形；微分球面定理；里奇曲率；里奇流；Submanifold； Differentiable Sphere Theorem； Ricci Curvature； Ricci Flow
摘要: 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。
Abstract: 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.
文章引用：常小芳. 欧氏空间子流形的一个微分球面定理[J]. 理论数学, 2025, 15(5): 85-89. https://doi.org/10.12677/pm.2025.155156

参考文献

[1] Hamilton, R.S. (1982) Three-Manifolds with Positive Ricci Curvature. Journal of Differential Geometry, 17, 255-306. [Google Scholar] [CrossRef] 
[2] Brendle, S. and Schoen, R. (2008) Manifolds with 1/4-Pinched Curvature Are Space Forms. Journal of the American Mathematical Society, 22, 287-307. [Google Scholar] [CrossRef] 
[3] Brendle, S. and Schoen, R.M. (2008) Classification of Manifolds with Weakly 1/4-Pinched Curvatures. Acta Mathematica, 200, 1-13. [Google Scholar] [CrossRef] 
[4] Xu, H. and Zhao, E. (2009) Topological and Differentiable Sphere Theorems for Complete Submanifolds. Communications in Analysis and Geometry, 17, 565-585. [Google Scholar] [CrossRef] 
[5] Brendle, S. (2008) A General Convergence Result for the Ricci Flow in Higher Dimensions. Duke Mathematical Journal, 145, 585-601. [Google Scholar] [CrossRef] 
[6] Xu, H. and Tian, L. (2011) A Differentiable Sphere Theorem Inspired by Rigidity of Minimal Submanifolds. Pacific Journal of Mathematics, 254, 499-510. [Google Scholar] [CrossRef] 
[7] Xu, H. and Gu, J. (2013) Geometric, Topological and Differentiable Rigidity of Submanifolds in Space Forms. Geometric and Functional Analysis, 23, 1684-1703. [Google Scholar] [CrossRef] 

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