C 1 Isometric Imbeddings

Isometric Embeddings of Pro-Euclidean Spaces

“…Then using Theorem 1 it can be seen that M is a proEuclidean space of rank at most n. Our Main Theorem then is a partial generalization of (the C version of) the famous Nash isometric embedding theorem found in [10] which states: every n-dimensional Riemannian manifold admits a C isometric embedding into E n+ . …”

“…The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E n+ . Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C version of) the famous Nash isometric embedding theorem from [10]. …”

“…To see that the sequence (ε i ) can be chosen so that the sequence (h i ) converges to an embedding we use a technique similar to one used by Nash in [10]. Consider the collection of sets…”

“…The most famous results in this area of mathematics are the Nash isometric embedding theorems for Riemannian manifolds [10] and [11]. One could then try to use these results in the limiting idea mentioned above, and indeed both Petrunin [12] and Le Donne [6] observed that one can use the C Nash isometric embedding theorem to prove that every n-manifold equipped with a sub-Riemannian metric admits an isometric embedding into E n+ .…”

Nash Twist and Gaussian Noise Measure for Isometric C1 Maps

“…Our point of interest lies in the space of isometric immersions of a Riemannian manifold (M, g) into a Euclidean space R q with the canonical metric h. In 1954, Nash proved that if a manifold M with a Riemannian metric g can be embdedded in a Euclidean space R q , q > n+1, then one can construct a large class of isometric C 1 embeddings ( [9]). If the initial embedding f 0 : M → R q is strictly short, that is if g − f * h is a Riemannian metric on M then the isometric embeddings can be made to lie in an arbitrary C 0 neighbourhood of the initial embedding.…”