Canonical Expressions of Algebraic Curvature Tensors
Canonical Expressions of Algebraic Curvature Tensors
复制标题
代数曲率张量的规范表达式
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Kaitlin Ragosta
中科院分区：
文献类型：
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作者：
Kaitlin Ragosta
代数曲率张量可以用多种方式表示，并且开发可以区分它们的不变量是有帮助的。一个潜在的不变量是R的签名，它可以以多种方式定义，类似于内积的签名。本文证明了定义在dim（V）= n的向量空间V上的任何代数曲率张量都可以仅用来自秩为k或更高的形式的正则代数曲率张量来表示，对于{2，…，n}，并且这样的表达式不是唯一的，消除了人们可能定义R的签名的一些可能性。我们还提供了表示任何给定R所需的秩为k的代数曲率张量的最小数目的界。
Algebraic curvature tensors can be expressed in a variety of ways, and it is helpful to develop invariants that can distinguish between them. One potential invariant is the signature of R, which could be defined in a number of ways, similar to the signature of an inner product. This paper shows that any algebraic curvature tensor defined on a vector space V with dim(V) = n can be expressed using only canonical algebraic curvature tensors from forms with rank k or higher for any k in {2,...,n}, and that such an expression is not unique, eliminating some possibilities for what one might define the signature of R to be. We also provide bounds on the minimum number of algebraic curvature tensors of rank k needed to express any given R.