Nash embedding theorem

The embedding theorem of Nash (after John Forbes Nash Jr.) is a result from the mathematical branch of Riemannian geometry. He says that every Riemannian manifold can be isometrically embedded in a Euclidean space of a suitable. " Isometric " is meant in the sense of Riemannian geometry: get the lengths of tangent vectors and the lengths of curves remain in the manifold. The usual Euclidian metric of the predetermined metric should induce the Riemannian manifold embedded in the sub- manifold, such that in the local coordinate of the Embedding:

Riemannian manifolds can be thought of as submanifolds of a Euclidean space so always. The dimension of the Euclidean space is generally, however, significantly greater than that of the Riemannian manifold.

The analogous result for ordinary differentiable manifolds is the embedding theorem of Whitney, which is much simpler in nature.

An embedding in the local real analytic case was proved in 1926 by Elie Cartan and Maurice Janet ( with, where the dimension of the Riemannian manifold is ). Nash proved the feasibility of the global embedding first for differentiable embeddings ( improved by Nicolaas Kuiper ), then in the case. In the global real analytic case Nash gave a proof in 1966.

The proof of Nash has been simplified by 1989 Matthias Günther ( University of Leipzig).

This results in each case bounds for the height of the dimension of the function of the dimension of the embedded Riemannian manifold, for example, in the case of Nash and Kuiper. In case ( ) Nash showed in 1956 the existence of a global embedding for ( compact manifold ), or ( non- compact case).

In his work of 1956 Nash laid the foundations for the Nash -Moser technique often found application in the theory of nonlinear partial differential equations.