Collapsing 4-manifolds under a lower curvature bound
Collapsing 4-manifolds under a lower curvature bound
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曲率下限下折叠 4 流形
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发表时间：
2012
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通讯作者：
Takao Yamaguchi
中科院分区：
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作者：
Takao Yamaguchi
本文描述了具有截面曲率一致下界和直径一致上界的四维闭合可定向黎曼流形的拓扑结构，它们坍缩为低维度量空间。这使我们能够理解具有曲率和直径几何界的闭可定向4流形的同胚类集。在证明上述结果的过程中，我们得到了具有非负曲率的四维完全非紧Alexandrov空间的灵魂定理。给出了具有非负曲率的三维完全Alexandrov空间的度量分类。
In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orientable 4-manifolds with those geometric bounds on curvature and diameter. In the course of the proof of the above results, we obtain the soul theorem for 4-dimensional complete noncompact Alexandrov spaces with nonnegative curvature. A metric classification for 3-dimensional complete Alexandrov spaces with nonnegative curvature is also given.