An investigation of several physical and mathematical issues concerning the structure of Einstein's General Theory of Relativity, (GR) will be undertaken. GR can be understood by its analogy with the more familiar ideas of electricity and magnetism. Most often it has been useful to study gravitational phenomena as just another classical field and use the analogies with electro-magnetic theory for the understanding of different physical phenomena or as a guide to the problems of a quantum theory of gravity. Recently, there has been developed a new point of view towards general relativity, where the analogy with electro- magnetic fields completely breaks down. The primary variables are no longer conventional fields but are instead non-local structures; in particular, they are three dimensional surfaces in space-time with the special property that light can travel along these surfaces, i.e., the surfaces are null or characteristic surfaces. It has been possible (though it was not at all obvious how to do so) to give a formulation of GR where the basic equations are for the determination of these surfaces. In turn, when these surfaces are determined the geometry of the space-time turns out to be identical with the conventional view derived from Einstein's general relativity. In other words, general relativity has been reformulated as a theory of three dimensional surfaces. The major portion of this project involves the study of the properties of these surface equations. We have been interested in both their classical aspects - (their geometric meaning, their consequences and methods of integration and approximation) and what relationship they might have with the difficulties associated with the construction of a quantum theory of gravity.
