THE LINEAR INDEPENDENCE OF SETS OF TWO AND THREE CANONICAL ALGEBRAIC CURVATURE TENSORS
THE LINEAR INDEPENDENCE OF SETS OF TWO AND THREE CANONICAL ALGEBRAIC CURVATURE TENSORS
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两个和三个正则代数曲率张量集的线性独立性
DOI：
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发表时间：
2010
期刊：
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通讯作者：
C. Dunn
中科院分区：
文献类型：
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作者：
A. Díaz;C. Dunn
将向量空间自伴自同态构造正则代数曲率张量的方法推广到任意自同态。如果满足一定的基本秩要求，我们建立了一个经典事实的匡威：如果A是对称的，那么RA是代数曲率张量。这使我们能够建立一个同时对角化的结果，在事件中，三个代数曲率张量是线性相关的。我们利用这些结果建立了一组两个或三个代数曲率张量线性无关的充分必要条件。我们用初等方法证明了这些结果。
We generalize the construction of canonical algebraic curvature tensors by self- adjoint endomorphisms of a vector space to arbitrary endomorphisms. Provided certain basic rank requirements are met, we establish a converse of the classical fact that if A is symmetric, then RA is an algebraic curvature tensor. This allows us to establish a simultaneous diagonalization result in the event that three algebraic curvature tensors are linearly dependent. We use these results to establish necessary and sufficient conditions that a set of two or three algebraic curvature tensors be linearly independent. We present the proofs of these results using elementary methods.