A note on closed isometric embeddings

A famous theorem due to Nash ([3]) assures that every Riemannian manifold can be embedded isometrically into some Euclidean space E. An interesting question is whether for a complete manifold M we can find a closed isometric embedding. This note gives the affirmative answer to this question asked to the author by Paolo Piccione. In his famous 1956 article John Nash proved that every Riemannian metric on an ndimensional manifold M can be constructed as a pullback metric for an embedding of M into some Euclidean space. He gave also an estimate of the smallest possible dimension N of the Euclidean space as N = 1 2 · n · (n + 1) · (3n + 11). Now one can try to find some stronger derivates of this theorem if strengthening the assumptions. In this note, we want to examine the question whether every complete manifold admits a closed isometric embedding. Although folk wisdom apparently has a positive answer to this question already, there does not seem to be any proof in the literature up to now. The question is more difficult than it might seem at first sight as there are plenty of non-closed isometric embeddings of complete manifols, e.g. spirals converging to 0 or to a circle as isometric embeddings of R. For a Lipschitz function f on a metric spaceM , we denote by L(f) := sup{ |f(p)−f(q) d(p,q) | p, q ∈ M,p 6= q} its Lipschitz number. We will need the following nice theorem about approximation of Lipschitz functions by smooth functions from [2] (Proposition 2.1 and its corollary): Theorem 1 Let (M, g) be a finite-dimensional Riemannian manifold, let f : M → R be a Lipschitz function, let ρ, r > 0. Then there is a C∞ and Lipschitz function g : M → R with |f(p)− g(p)| ≤ ρ for every p ∈M , and L(g) ≤ L(f) + r. Interestingly, this theorem even has a refinement on infinite-dimensional separable Riemannian manifolds, cf. [1]. Now let us state and prove our theorem. The basic idea of the proof is to look at balls of increasing radius and to define an imbedding which lifts the larger and larger balls into an additional direction thereby resolving a possible spiralling. As the distance itself is not differentiable in general, we have to be a little bit more careful and thus we will need the theorem above. Theorem 2 If (M, g) is a complete n-dimensional Riemannian manifold, then there is a closed isometric C∞-embedding of (M, g) into E, where N := 12 · n · (n+ 1) · (3n+ 11). ∗Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM) Campus Morelia, C. P. 58190, Morelia, Michoacán, Mexico. email: [email protected]