Gromoll-Meyer球面上的一个正曲率度量
A positively curved metric on the Gromoll-Meyer sphere
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中文总结 AI 辅助
该文在Sp(2)上构造双不变度量的显式形变，使Gromoll-Meyer exotic 7-球面在充分小正参数下获得正截面曲率，证明结合均匀估计与代数分析，并给出临界集上三次系数的正下界。
中文摘要 AI 辅助
我们在Sp(2)上构造了双不变度量的一个显式形变，使得该形变在Gromoll-Meyer exotic 7-球面上诱导的度量，对所有充分小的正参数都具有正截面曲率。该形变由一个与商作用相容的固定自伴算子决定。证明结合了均匀曲率估计与对水平交换平面的代数分析。在低阶正项消失的临界集上，我们建立了曲率分子三次系数的显式正下界。对四阶余项的均匀控制以及一个紧致性论证表明，每个截面曲率均为正。
英文摘要
We construct an explicit deformation of the bi-invariant metric on Sp(2) whose induced metrics on the Gromoll-Meyer exotic 7-sphere have positive sectional curvature for all sufficiently small positive parameters. The deformation is determined by a fixed self-adjoint operator compatible with the quotient action. The proof combines a uniform curvature estimate with an algebraic analysis of the horizontal commuting planes. On the critical set where the lower-order positive terms vanish, we establish an explicit positive lower bound for the cubic coefficient of the curvature numerator. Uniform control of the fourth-order remainder and a compactness argument then show that every sectional curvature is positive
