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

We study the problem of construction of explicit isometric embeddings of (pseudo)-Riemannian manifolds. We discuss the method which is based in the idea that the exterior symmetry of the embedded surface and the interior symmetry of the metric on it must be the same. In case of high enough symmetry of the metric such method allows to transform the metric inducedness condition, 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 of use in the construction of embeddings for metrics whose symmetry group is hard to analyze, as well for the construction of isometrically deformed (bended) surface. We give some examples of application of this method. In particular, we construct the embedding of spatially-flat Friedmann model and isometric bendings of sphere, 3-sphere and squashed AdS universe, which is connected to the Godel universe.