Fundamental groups of manifolds with positive isotropic curvature

• 成果类型: Article
• 署名作者: Fraser, AM
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2003.158.345
• 发表日期: 2003
• 页码: 345-354
• 关键词: SCALAR CURVATURE; minimal immersions; euclidean-space; SURFACES; submanifolds; 4-manifolds; EXISTENCE; 2-spheres; 2-planes; TOPOLOGY

摘要

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. By the results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC: namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds with PIC. An important open problem has been to understand the fundamental groups of manifolds with PIC. In this paper we prove a new result in this direction. We show that the fundamental group of a compact manifold M-n with PIC, n greater than or equal to 5, does not contain a subgroup isomorphic to Z circle plus Z. The techniques used involve minimal surfaces.