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Structure Groups of Algebraic Curvature Tensors in Dimension Three

The purpose of this paper is to examine structure groups of algebraic curvature tensors in dimension three. In dimension two, the structure group of algebraic curvature tensors is known to always be SL2 . In dimension three, there is more than one non-zero algebraic curvature tensor, up to multiple, which complicates the task. The two main goals used in this paper were, first, to consider an invertible 3x3 matrix A, one of each Jordan Normal form and find the subspace of algebraic curvature tensors preserved by A, and second, to classify the Lie algebras of the three non-trivial cases of Lie algebras of the structure groups of algebraic curvature tensors in dimension three which are known. The idea is to classify these Lie algebras and then given an element A, it is possible to see if it is in a structure group of some algebraic curvature tensor. In particular, in the cases where the subspace of curvature tensors preserved by a given matrix was of dimension one, the algebraic curvature tensor that spans that subspace is found. The general question is, given a subgroup H of GLn, does there exist an algebraic curvature tensor R such that its structure group is H.