Double forms, curvature structures and the $(p,q)$-curvatures
Double forms, curvature structures and the $(p,q)$-curvatures
复制标题
双精度形式、曲率结构和 $(p,q)$-曲率
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Labbi
中科院分区：
文献类型：
--
作者：
M. Labbi
我们将度量张量和Hodge星算子引入到二重形式代数中，以研究该代数结构的一些方面。这些性质随后被用于研究新的黎曼曲率不变量，称为(p, q)-曲率。它们是通过将高斯-克罗内克张量替换为黎曼曲率张量得到的p曲率的推广。特别地，对于p = 0, （0, q）曲率与H. Weyl曲率不变量重合，对于p = 1, （1, q）曲率是广义爱因斯坦张量的曲率，对于q = 1, （p, 1）曲率与p曲率重合。此外，我们还证明了第二H. Weyl曲率不变量对于维数为nbbbb4的爱因斯坦流形是非负的，对于零标量曲率的共形平坦流形是非正的。对于高H. Weyl曲率不变量也证明了类似的结果。
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.