IMMERSION IDEALS AND THE CAUSAL-STRUCTURE OF RICCI-FLAT GEOMETRIES

A moving-frame analysis is given for the immersion of four-dimensional (pseudo-) Riemannian geometries in ten-dimensional (pseudo-) Euclidean space. The resulting Darboux bundles incorporate auxiliary connection forms that give concise quadratic expressions for the Riemann and Ricci tensors. Using Cartan-Kahler theory we calculate Cartan characters for the Ricci-flat case, which directly show how the three-dimensional foliation, constraints and causal evolution of dynamic geometry arise. Cartan character analyses are also reported for three-dimensional flat Riemannian geometry immersed in six-dimensional Euclidean space, for five-dimensional Ricci-flat geometry immersed in 15, and for some coupled and specialized fields. In all cases (m - 1)-dimensional causal foliation naturally emerges from use of the auxiliary variables describing the immersions, and the physical degrees of freedom are immediately discovered.