Explicit isometric embeddings of pseudo-Riemannian manifolds: ideas and applications

Abstract

We study the problem of construction of explicit isometric embeddings of (pseudo)-Riemannian manifolds. We discuss the method, which is based on the idea that the exterior symmetry of the embedded surface and the interior symmetry of its metric must be the same. In case of high enough symmetry of the metric such method allows transforming the expression for induced metric, which is the one to be solved in order to construct an embedding, into a system of ODEs. It turns out that this method can be generalized to allow the surface to have lower symmetry as long as the above simplification occurs. This generalization can be used in the construction of embeddings for metrics, whose symmetry group is hard to analyze, and the construction of the isometrically deformed (bent) surface. We give some examples of the application of this method. In particular, we construct the embedding of spatially-flat Friedmann model and isometric bendings of a sphere, 3-sphere, and squashed AdS universe, which is related to the Godel universe.