Invariant classification and equivalence in general relativity

Einstein's theory of general relativity is one of the two great contributions of modern mathematics to theoretical physics. In effect, general relativity _is_ geometry, the abstract study of non-Euclidean curved space. The physical properties that govern the motion of stars, planets, galaxies indeed the entire universe arise from an analysis of such geometry.Historically, the mathematics of curved abstract spaces (manifolds) anticipated the application to physics. The initial formulation is due to Bernhard Riemann who work in the latter half of the 19th century, and by mathematicians such as Sophus Lie and Elie Cartan. Indeed, the two disciplines --- mathematics on the one hand and the theoretical physics of general relativity --- develop in a parallel, intertwined fashion. I hope to be able to make a contribution by developing certain mathematical aspects of the theory.