Canonical Algebraic Curvature Tensors of Symmetric and Anti-Symmetric Builds

Abstract

We relate canonical algebraic curvature tensors that are built from a self-adjoint ($R^S_A$) or skew adjoint ($R^{\Lambda}_A$) linear operator A. Several authors have proven that any algebraic curvature tensor $R$ may be expressed as a sum of $R^S_A$, or as a sum of $R^{\Lambda}_A$. This motivates our interest in relating them as well as in the linear independence of sets of canonical algebraic curvature tensors. We develop an identity that relates $R^{\Lambda}_A$ to $R^S_A$, which will allow us to employ previous methods used for $R^S_A$ to the case of $R^{\Lambda}_A$ as well as use them interchangeably in some instances. We compute the structure group of $R^{\Lambda}_A$, and develop methods for determining the linear independence of sets which contain both $R^{\Lambda}_A$ and $R^S_A$. We consider cases where the operators are arranged in chain complexes and find that this greatly restricts the linear independence of the curvature tensors with those operators. Moreover, if one of the operators has a nontrivial kernel, we develop a method for reducing the bound on the least number of canonical algebraic curvature tensors that it takes to write a canonical algebraic curvature tensor.