In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates.
Given an m-dimensional Riemannian manifold (M, g), an isometric embedding is a continuously differentiable topological embedding f: M → ℝn such that the pullback of the Euclidean metric equals g. In analytical terms, this may be viewed (relative to a smooth coordinate chart x) as a system of 1/2m(m + 1) many first-order partial differential equations for n unknown (real-valued) functions: If n is less than 1/2m(m + 1), then there are more equations than unknowns.
Then there is a sequence of continuously differentiable isometric embeddings (or immersions) M → ℝn of g which converge uniformly to f.The theorem was originally proved by John Nash with the stronger assumption n ≥ m + 2.
[7] Furthermore, some smooth isometric embeddings exhibit rigidity phenomena which are violated by the largely unrestricted choice of f in the Nash–Kuiper theorem.
Any such embedding can be scaled by an arbitrarily small constant so as to become short, relative to any given Riemannian metric on the surface.
It follows from the Nash–Kuiper theorem that there are continuously differentiable isometric embeddings of any such Riemannian surface where the radius of a circumscribed ball is arbitrarily small.
Although Whitney's theorem also applies to noncompact manifolds, such embeddings cannot simply be scaled by a small constant so as to become short.
Nash proved that every m-dimensional Riemannian manifold admits a continuously differentiable isometric embedding into ℝ2m + 1.
However, Nash's method of proof was adapted by Camillo De Lellis and László Székelyhidi to construct low-regularity solutions, with prescribed kinetic energy, of the Euler equations from the mathematical study of fluid mechanics.
The basic idea in the proof of Nash's implicit function theorem is the use of Newton's method to construct solutions.