发表机构
Zhejiang Normal University; University of California, San Diego（浙江师范大学; 加利福尼亚大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过单一强归纳法，利用平坦子空间、根中心化子及迹消失引理，给出了 Simons 黎曼和乐定理的完整代数证明，并推导了流形的局部对称性。
AI 中文摘要
我们通过关于曲率轨道线性张成空间维数的单一强归纳，给出了 Simons 黎曼和乐定理的一个详细且完整的代数证明。该证明使用了平坦子空间和根中心化子的构造，但用迹消失引理取代了对公共零权空间的详细分析，后者是先前代数证明中最具技术性的核心部分。该引理适用于包含在湮灭曲率模的理想中的曲率算子。限制到公共测地子空间会产生一个适当的不变核。归纳使该核在和乐作用下固定，之后迹引理表明根中心化子张成整个环境空间。它们的交集恢复极大平坦子空间，不可约性完成证明。为了完整性，我们还从代数定理出发，详细推导了底层黎曼流形的局部对称性。
英文摘要
We give a detailed and complete algebraic proof of Simons' Riemannian holonomy theorem by a single strong induction on the dimension of the linear span of the curvature orbit. The proof uses the construction of flats and root centralizers, but replaces the detailed analysis of the common zero-weight space, the most technical core of previous algebraic proofs, by a trace-vanishing lemma. The lemma applies to curvature operators contained in an ideal that annihilates the curvature module. Restriction to a common geodesic subspace produces a proper invariant kernel. Induction makes this kernel fixed by the holonomy action, after which the trace lemma shows that the root centralizers span the ambient space. Their intersection recovers the maximal flat, and irreducibility completes the proof. For the completeness we also include the detailed derivation of the local symmetry of the underlying Riemannian manifold from the algebraic theorem.
