Isometric embeddings of Riemannian and pseudo-Riemannian manifolds.

Evolution of Density Parameters on a Smooth Embedded Universe

“…(3) and (4) are consequences of the embedding with a regular and differentiable map X : V 4 → V 5 , where V 4 and V 5 are Riemannian geometries. The main result of the Nash embedding theorem, [25] latter extended to pseudoRiemannian geometries, [40] shows that any n-dimensional Riemannian geometry V n can be obtained by a sequence of infinitesimal perturbations of a non-perturbed metric g μν given by the non-perturbed extrinsic curvature tensor k μν as follows: Suppose we have an embedded Riemannian manifold with metric g μν . Then, another Riemannian geometry with metric g μν + δg μν can be generated by…”

Recent progress on the notion of global hyperbolicity

“…The existence of a steep temporal function is interesting for a spacetime M , as it solves the problem of its isometric embeddability in some L N (in the spirit of Nash' theorem [36]), see [35]. In fact, it was noticed by Greene [25] and Clarke [14] that any semi-Riemannian (or even degenerate) manifold can be isometrically immersed in some semi-Euclidean space R N s of sufficiently big dimension N and index s. The problem is a bit subtler for s = 1, but a simple argument in [35] shows that a spacetime can be isometrically embedded in L N if and only if it admits a steep temporal function.…”

Relative isometric embeddings of Riemannian manifolds

“…Our proofs of Theorems 1.1 and 1.2, which are presented in Section 3, employ a variation of a Cartesian product technique used in Nash's work [15,Part D], see also [5], to piece together the desired isometric embedding from certain other maps. One of these maps, which we construct in Section 2, is a short mapping f 1 : M → R m which sends most of M to the origin, preserves f on a neighborhood V of p, and is strictly short everywhere else.…”