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Extra info for Communications in Mathematical Physics - Volume 257
64]. The first question was answered affirmatively in , and our aim is to answer the other two. 1. Let (M, g) be a globally hyperbolic spacetime. Then, it is isommetric to the smooth product manifold R × S, ·, · = −β dT 2 + g, ¯ The second-named author has been partially supported by a MCyT-FEDER Grant, MTM200404934-C04-01. 1 See also the authors’ contribution to Proc. II Int. Meeting on Lorentzian Geometry, Murcia, Spain, 2003, Publ. RSME vol. 8 (2004) 3–14, gr-qc/0404084. N. Bernal, M. S´anchez where S is a smooth spacelike Cauchy hypersurface, T : R × S → R is the natural projection, β : R × S → (0, ∞) a smooth function, and g¯ a 2-covariant symmetric tensor field on R × S, satisfying: 1.
Cambridge Monographs on Mathematical Physics, No. 1. London-NewYork: Cambridge University Press, 1973 6. : General Relativity and Cosmology. Bull. Amer. Math. Soc. 83(6), 1101–1164 (1977) 7. : Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch’s splitting. A revision. In: Proceedings of the 13th School of Differential Geometry, Sao Paulo, Brazil, 2004 (to appear in Matematica Contemporanea). W. Gibbons Commun. Math. Phys. 1007/s00220-004-1260-y Communications in Mathematical Physics Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators Guillaume James, Yannick Sire Math´ematiques pour l’Industrie et la Physique, UMR CNRS 5640, and D´epartement GMM, Institut National des Sciences Appliqu´ees, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France.
Int. J. Math. Sci. no. 7, 405–450 (2003) 21. : Spacetime symmetries and linearization stability of the Einstein equations I. J. Math. Phys. 16, 493–498 (1975) 22. Witt, D. : Vacuum space-times that admit no maximal slice. Phys. Rev. Lett. W. Gibbons Commun. Math. Phys. 1007/s00220-005-1346-1 Communications in Mathematical Physics Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes Antonio N. Bernal, Miguel S´anchez Dpto. E. Ehrlich, wishing him a continued recovery and good health Abstract: The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function T whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting M = R × S, g = −β(T , x)dT 2 + g¯ T , (b) if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function T with timelike gradient ∇T on all M.