正截面曲率的奇异$8$-球面与三阶同伦$10$-球面
Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
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中文总结 AI 辅助
本文证明奇异$8$-球面及三阶同伦$10$-球面存在严格正截面曲率度量，结合Sperança丛模型与He-Liu-Yau构造，利用Reiser-Wraith粘合定理完成证明。
中文摘要 AI 辅助
我们证明了奇异光滑$8$-球面以及代表三阶元素的两个定向同伦$10$-球面均允许具有严格正截面曲率的黎曼度量。该构造结合了Sperança的特殊$S^3$-$S^3$丛模型与He、Liu和Yau的相容圆盘构造。我们证明了相关丛允许等变极坐标正规形，其转移函数沿子午线为常数且共轭等变。我们还证明了He--Liu--Yau构造中依赖于表示的部分仅需对无穷小作用场的一致有界性。Sperança的$8$维和$10$维例子满足此界。所得的北部和南部度量具有匹配的边界度量及相容的第二基本形式，因此Reiser--Wraith粘合定理给出了所需的正曲率度量。
英文摘要
We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He--Liu--Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser--Wraith gluing theorem gives the required positively curved metrics.
发表机构
- Durham University（杜伦大学）
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