第二类曲率算子刚性条件的新结果
New Curvature Operator of the Second Kind Conditions for Rigidity
- Auburn University（奥本大学）
- University of California, Los Angeles（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过第二类曲率算子建立Tachibana型刚性定理，证明维数≥5的完备Einstein流形在$C_n$-非负条件下具有常非负截面曲率，并推广至平移算子与对称空间分类。
AI中文摘要：
我们借助第二类曲率算子（COSK）建立了Tachibana型刚性定理。一个结合Weyl张量上对称与交换子作用的尖锐估计导出一个常数$C_{n}$，其中$\frac{C_{n}}{n}\to\frac{5}{4}$，使得任意维数$n\geq5$且具有$C_{n}$-非负COSK的完备Einstein流形具有常非负截面曲率。我们还探讨了第二类平移曲率算子，得到类似的刚性结果，为调和形式提供了一个优雅的Ricci修正Bochner恒等式，并在所得平移锥中分类了完备单连通局部对称Einstein流形。显式空间证明了张量估计的尖锐性，对称空间COSK特征值的完整列表识别了其中的临界情形。
英文摘要：
We establish Tachibana-type rigidity theorems with the aid of the curvature operator of the second kind (COSK). A sharp estimate that combines the symmetric and commutator actions on Weyl tensors leads to a constant $C_{n}$, with $\frac{C_{n}}{n}\to\frac{5}{4}$, such that a complete Einstein manifold of dimension $n\geq5$ with $C_{n}$-nonnegative COSK has constant nonnegative sectional curvature. We also explore shifted curvature operators of the second kind leading to similar rigidity results, offer an elegant Ricci-corrected Bochner identity for harmonic forms, and classify the complete simply connected locally symmetric Einstein manifolds in the resulting shifted cone. Explicit spaces establish the sharpness of the tensor estimates, and complete lists of the eigenvalues of the COSKs for symmetric spaces identify which of them are borderline cases.
