Canonical Expressions of Algebraic Curvature Tensors

Abstract

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.