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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lorentzian manifolds isometrically embeddable in $ \mathbb{L}^N$


Authors: O. Müller and M. Sánchez
Journal: Trans. Amer. Math. Soc. 363 (2011), 5367-5379
MSC (2010): Primary 53C50, 53C12, 83E15, 83C20
Published electronically: May 12, 2011
MathSciNet review: 2813419
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Abstract: In this article, the Lorentzian manifolds isometrically embeddable in $ \mathbb{L}^N$ (for some large $ 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 $ \tau$ with $ \vert\nabla \tau\vert>1$. Then, we prove that any globally hyperbolic spacetime $ (M,g)$ admits such a function, and, even more, a global orthogonal decomposition $ M=\mathbb{R} \times S, g=-\beta dt^2 + g_t$ with bounded function $ \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 $ \mathbb{L}^N$.

The role of the so-called ``folk problems on smoothability'' in Clarke's approach is also discussed.


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Additional Information

O. Müller
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, C. P. 58190 Morelia, Michoacán, México
Address at time of publication: Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, D-93059 Regensburg, Germany
Email: olaf@matmor.unam.mx, Olaf.Mueller@mathematik.uni-regensburg.de

M. Sánchez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain
Email: sanchezm@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05299-2
PII: S 0002-9947(2011)05299-2
Keywords: Causality theory, globally hyperbolic, isometric embedding, conformal embedding
Received by editor(s): July 19, 2009
Received by editor(s) in revised form: December 18, 2009
Published electronically: May 12, 2011
Article copyright: © Copyright 2011 American Mathematical Society