The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rⁿ. John Forbes Nash John Forbes Nash Jr. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

"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 C^k, 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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, an analytic function is one that is locally given by a convergent power series. ...

The C¹ theorem was published in 1954, the C^k-theorem in 1956, and the analytical case was done in 1966 by John Nash. See h-principle for further developments. John Forbes Nash John Forbes Nash Jr. ... The homotopy principle (h-principle) is a very general way to solve partial differential equations PDE (and more generally partial differential relations PDR). ...

Nash-Kuiper theorem (C¹ embedding theorem)

Theorem. Let (M,g) be a Riemannian manifold and is a short smooth embedding (or immersion) into Euclidean space Eⁿ, . Then for arbitrary ε > 0 there is an embedding (or immersion) which is In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

(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) − f_ε(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 Euclidean space. The theorem was originally proved by J. Nash with condition instead of and generalized by Nicolaas Kuiper, by a relatively easy trick. In differential topology, the Whitney embedding theorem states that Any smooth second-countable -dimensional manifold can be embedded in Euclidean -space. ...

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 (from Gauss formula, there is no such C²-embedding). A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

C^k embedding theorem

The technical statement is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C^k, 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 C^k) such that for every point p of M, the derivative df_p is a linear map from the tangent space T_pM to Rⁿ which is compatible with the given inner product on T_pM and the standard dot product of Rⁿ in the following sense: In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In 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. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...

< u, v > = df_p(u) · df_p(v)

for all vectors u, v in T_pM. This is an undetermined system of partial differential equations (PDE's). In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...

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). In 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. ... In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... This article is about the mathematical concept of convolution. ... 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 existential quantifier. ...

References

• N.H.Kuiper: "On C¹-isometric imbeddings I", Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 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.