Olaf Müller

This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?

In this article, the Lorentzian manifolds isometrically embeddable in L N \mathbb {L}^N (for some large N N , in the spirit of Nash’s theorem) are characterized as a subclass of the set of all stably causal spacetimes; concretely, those which admit a smooth time function τ \tau with | ∇ τ | > 1 |\nabla \tau |>1 . Then, we prove that any globally hyperbolic spacetime ( M , g ) (M,g) admits such a function, and, even more, a global orthogonal decomposition M = R × S , g = − β d t 2 + g t M=\mathbb {R} \times S, g=-\beta dt^2 + g_t with bounded function β \beta and Cauchy slices. In particular, a proof of a result stated by C.J.S. Clarke is obtained: any globally hyperbolic spacetime can be isometrically embedded in Minkowski spacetime L N \mathbb {L}^N . The role of the so-called “folk problems on smoothability” in Clarke’s approach is also discussed.

We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bounded away from zero -and, thus, an orthogonal global splitting of the spacetime with bounded lapse. The role of the so-called "folk problems on smoothability" is stressed.

The aim of this article is to present the category of bounded Frechet manifolds in respect to which we will review the geometry of Frechet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some applications to Riemannian geometry are given.