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"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.
The C1 theorem was published in 1954, the Ck-theorem in 1956, and the analytical case was done in 1966 by John Nash. See h-principle for further developments.
Theorem. Let be a Riemannian manifold and is a short smooth embedding (or immersion) into Euclidean space , . Then for arbitrary there is an embedding (or immersion) which is
In particular, as it follows from Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric -embedding in 2m-dimensional Euclidean 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 embedded into an arbitrarily small ball in Euclidean 3-space (clearly there is no such -embedding).
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 productIn mathematics, an inner product space is a vector space with additional structure, an inner product scalar product or dot product which allows us to introduce geometrical notions such as angles and lengths of vectors. Inner product spaces are generalizat on TpM and the standard dot product of Rn in the following sense:
for all vectors u, v in TpM. This is an undetermined system of partial differential equationIn mathematics, and in particular calculus, a partial differential equation PDE is an equation involving partial derivatives of an unknown function. The idea is to describe a function indirectly by a relation between itself and its partial derivatives, ras (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 theoremIn mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as a function of the others. More precisely, that if f R m ''n rarr R n a is in R m a 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 methodIn numerical analysis, Newton's method (or the Newton-Raphson method is an efficient algorithm for finding approximations to the zero (or root) of a real-valued function. As such, it is an example of a root-finding algorithm. It can also be used to find t 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 convolutionIn mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version. The smoothing operators ensure that the iteration converges to a root and so it can be used as an existence theoremProofs In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s). or more generally 'for all x y . there exist(s). That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existenti 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).