Rigidity of Einstein 4-manifolds with positive curvature
Rigidity of Einstein 4-manifolds with positive curvature
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DOI：
10.1007/pl00005792
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发表时间：
2000-11
影响因子：
3.1
通讯作者：
Dagang Yang
中科院分区：
文献类型：
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作者：
Dagang Yang
4-流形上具有正数量曲率的爱因斯坦度量被称为正规化的IFRIC=1。黎曼几何中的一个基本问题是将具有正截面曲率的爱因斯坦4-流形归类为拓扑范畴、微分同胚范畴或等距范畴。本文证明了，如果规格化爱因斯坦4-流形M的截面曲率K满足下界K≥ε0，ε0≡(-23)/120≈0.102843，或定理1.1的条件(B)，则它与截面曲率K=1/3的S4，RP4或具有规格化Fubini-Study度量的CP2等距.因此，在S4和CP2上满足上述两个条件之一的爱因斯坦度量的归一化模空间都只包含一个点。特别地，如果一个光滑的4-流形与S4、Rp4或Cp2同胚，但不同于这三个流形中的任何一个，则它不能支持任何满足这两个条件之一的规范化爱因斯坦度量。
An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized ifRic=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvatureKof a normalized Einstein 4-manifoldMsatisfies the lower boundK≥ε0, ε0≡( -23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to eitherS4,RP4with constant sectional curvatureK=1/3, orCP2with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions onS4andCP2contain only a single point. In particular, ifMis a smooth 4-manifold which is homeomorphic to eitherS4,RP4, orCP2but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions.