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The Nash embedding theorems (or imbedding theorems), also called Fundamental Theorem of Riemannian geometry. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn.
"Isometrically" means "preserving lengths of curves". The result therefore means that any Riemannian manifold can be visualized as a submanifold of Euclidean space.
The first theorem is for C1-smooth embeddings and the second for analytic or of class Ck, 3 ≤ k ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and is very counterintuitive, while the proof of the second one is very technical but result is not at all surprising.
C1 theorem was published in 1954, Ck-theorem in 1956 and analitical case was done in 1966 by John Nash
Theorem. Let be a Riemannian manifold and is a strictly short smooth embedding (or immersion) into Euclidean space , . Then for arbitrary there is an embedding (or immersion) which is
The theorem has many counterintuitive
implications. For example it follows that any closed oriented surface can be embedded into an arbitrarily small ball in Euclidean 3-space (clearly there is no such -embedding).
The method used in the proof gives a lot of similar paradoxes, for example it can be shown that the two-dimensional sphere can be turned inside-out in the class of of isometric immersions, also any n-dimensional Riemannin manifold admits a path isometric map to Euclidean n-space (see Smale's paradox, h-principle and convex integration)
The technical statement is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class Ck, 3 ≤ k ≤ ∞), then there exists a number n ( will do) and an injective map f : M -> Rn (also analytic or of class Ck) such that for every point p of M, the derivative dfp is a linear map from the tangent space TpM to Rn which is compatible with the given inner product on TpM and the standard dot product of Rn in the following sense:
The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into Rn. A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus. The proof of the global embedding theorem as presented here relies on Nash's far-reaching generalization of the implicit function theorem, the Nash-Moser inverse function theorem and
Newton's method with postconditioning (see ref.). The basic idea of Nash to
solve the embedding problem was to use Newton's method to prove the system of
PDEs has a solution. The standard Newton method fails to converge
when applied to the system, so Nash uses smoothing operators to
ensure to make the Newton iteration converge this adapted Newton method
is called Newton method with postconditioning. The smoothing operators
are defined by convolution. The smoothing operators ensure that the
iteration converges to a root and so it can be used as an existence theorem as well. By showing that the systems of PDE's has a root proves
the existence of isometric embedding of Riemannian manifolds.
There is also a older iteration called the Kantovorich iteration that
is an existence theorem using only Newton's method (so no smoothing operators).Nash-Kuiper Theorem (C1 embedding theorem)
In particular, as it follows from Whitney Embedding Theorem, any m-dimensional Riemannian manifold admits an isometric -embedding in 2m-dimensional Eucledean space. The theorem was originally proved by J. Nash with condition instead of and generalized by N.H.Kuiper, by a relatively easy trick.Ck embedding theorem
for all vectors u, v in TpM.
This is an undetermined system of partial differential equations (PDE's).
