Structure Groups of Pseudo-Riemannian Algebraic Curvature Tensors

An algebraic curvature tensor is the full Riemann curvature tensor of a manifold restricted to a single point, call the point p, and a model space is the tangent space at p paired with the algebraic curvature tensor there. The structure group of such an object is the set of all linear transformation on the tangent space of p which preserve the tensor. Most of these concepts are developed starting with familar Linear Algebra and should be accesable to many undergraduates. We are able to greatly characterize the elements of the structure groups in question and produce general forms of their matrix representations. We also work with the direct sum decomposition of model spaces, and this plays a large role in describing the elements of the structure group. It is also of note that much of this research is done in the arbitrary signature case, which will be described in this presentation. Also we will develop the motivation for this work and some conjectures and opportunities for further research.