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The Nash embedding theorems (or imbedding theorems) 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 the result is not at all surprising.
C1 theorem was published in 1954, Ck-theorem in 1956 and analytical case was done in 1966 by John Nash. See h-principle for further developments.
Nash-Kuiper Theorem (C1 embedding theorem)
Theorem. Let (M,g) be a Riemannian manifold and is a strictly
short smooth embedding (or immersion) into Euclidean space En,
. Then for arbitrary ε > 0 there is an embedding (or immersion) which is
- (i) C1-smooth,
- (ii) isometric, i.e. for any two vectors in the tangent space at we
have that .
- (iii) ε-close to f, i.e. : | f(x) -
fε(x) | < ε for any .
In particular, as it follows from Whitney
embedding theorem, any m-dimensional Riemannian manifold admits an isometric C1-embedding in 2m-dimensional Eucledean space. The theorem was originally proved by
J. Nash with condition instead of and generalized by Nicolaas Kuiper, by a relatively
easy trick.
The theorem has many counterintuitive implications. For example it follows that any closed oriented surface can be
C1 embedded into an arbitrarily small ball
in Euclidean 3-space (clearly there is no such C2-embedding).
Ck embedding theorem
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 (n = m2 + 5m + 3 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:
- < u, v > = dfp(u) ·
dfp(v)
for all vectors u, v in TpM. This is an undetermined system of partial differential equations (PDE's).
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
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).
References
- N.H.Kuiper "On C1-isometric imbeddings" I.Proc.Koninkl.Nederl.Ak.Wet. A-58 pp.545-556
- John Nash: "C1-isometric imbeddings", Annals of Mathematics, 60 (1954), pp 383-396.
- John Nash: "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1956), pp 20-63.
- John Nash: "Analyticity of the solutions of implicit function problem with analytic data" Annals of Mathematics, 84 (1966),
pp 345-355.
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