曲率算子与标量曲率不变量
Curvature operators and scalar curvature invariants
摘要：
我们继续研究的问题，当一个伪黎曼流形可以局部特征在于其标量多项式曲率不变量（从黎曼张量和协变导数）。我们进一步利用对齐理论和双向量形式的Weyl算子在更高的维度，并介绍了重要的概念diagonalisability和（复）解析度量扩展。我们表明，如果存在一个分析度量扩展的任意维空间的任何签名的黎曼空间（欧几里德签名），那么该空间的特点是其标量曲率不变量。特别是，我们讨论了洛伦兹的情况下，中性签名的情况下，在四维更详细。
英文摘要：
We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of alignment theory and the bivector form of the Weyl operator in higher dimensions, and introduce the important notions of diagonalisability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterised by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
