  Nontrivial isometric embeddings for flat metrics (i.e., those which are not
just planes in the ambient space) can serve as useful tools in the description
of gravity in the embedding gravity approach. Such embeddings can additionally
be required to have the same symmetry as the metric. On the other hand, it is
possible to require the embedding to be unfolded so that the surface in the
ambient space would occupy the subspace of the maximum possible dimension. In
the weak gravitational field limit, such a requirement together with a large
enough dimension of the ambient space makes embedding gravity equivalent to
General Relativity, while at lower dimensions it guarantees the linearizability
of the equations of motion. We discuss symmetric embeddings for the metrics of
flat Euclidean three-dimensional space and Minkowski space. We propose the
method of sequential surface deformations for the construction of unfolded
embeddings. We use it to construct such embeddings of flat Euclidean
three-dimensional space and Minkowski space, which can be used to analyze the
equations of motion of embedding gravity.

General Relativity and Quantum Cosmology