Immersion ideals and the causal structure of Ricci-flat geometries

Abstract

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 the authors 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 physical degrees of freedom are immediately discovered.