Given a globally hyperbolic spacetime M , we show the existence of a smooth spacelike Cauchy hypersurface S and, thus, a global diffeomorphism between M and R × S. * The authors acknowledge Prof. P. Ehrlich's clarifying comments on Geroch's theorem and subsequent references.

To Professor P.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 +ḡ 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 .

The classical definition of global hyperbolicity for a spacetime (M, g) comprises two conditions: (A) compactness of the diamonds J + (p) ∩ J − (q), and (B) strong causality. Here we show that condition (B) can be replaced just by causality. In fact, we show first that the classical definition of causal simplicity (which impose to be distinguishing, apart from the closedness of J + (p), J − (q)) can be weakened in causal instead of distinguishing. So, the full consistency of the causal ladder (recently proved by the authors in a definitive way) yields directly the result.2000 MSC: Primary 53C50, Secondary 53C80, 83C75.

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M , have been solved. Here we give further results, applicable to several problems:(1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible.(2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S = T −1 (0) is constructed -thus, the spacetime splits orthogonally as R × S in a canonical way.Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but nonacausal). Concretely, we construct a smooth function τ : M → R such that the levels S t = τ −1 (t), t ∈ R satisfy: (i) S = S 0 , (ii) each S t is a (smooth) spacelike Cauchy hypersurface for any other t ∈ R\{0}. If S is also acausal then function τ becomes a time function, i.e., it is strictly increasing on any future-directed causal curve.

Prevalence of HBV and HCV infection in IBD is similar to that of the general population of reference and lower than that in previously published series. This fact, in addition to the lack of association with invasive procedures, suggests the existence of adequate preventive measures in centers attending to these patients. The low percentage of effective vaccination makes it mandatory to intensify B virus vaccination in IBD.

The following three geometrical structures on a manifold are studied in detail:Leibnizian: a non-vanishing 1-form Ω plus a Riemannian metric ·, · on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized.Galilean: Leibnizian structure endowed with an affine connection ∇ (gauge field) which parallelizes Ω and ·, · . Fixed any vector field of observers Z (Ω(Z) ≡ 1), an explicit Koszul-type formula which reconstruct bijectively all the possible ∇'s from the gravitational G := ∇ZZ and vorticity ω := (1/2) rot Z fields (plus eventually the torsion) is provided.Newtonian: Galilean structure with ·, · flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and ω ≡ 0). Classical concepts in Newtonian theory are revisited and discussed.

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Physical foundations for relativistic spacetimes are revisited, in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of Special Relativity and Classical Mechanics) are shown to correspond with a double linear approximation in the measurement of space and time. While General Relativity appears by dropping the first linearization, Finsler spacetimes appear by dropping the second one. The classical Ehlers-Pirani-Schild approach is carefully discussed and shown to be compatible with the Lorentz-Finsler case. The precise mathematical definition of Finsler spacetime is discussed by using the space of observers. Special care is taken in some issues such as: the fact that a Lorentz-Finsler metric would be physically measurable only on the causal directions for a cone structure, the implications for models of spacetimes of some apparently innocuous hypotheses on differentiability, or the possibilities of measurement of a varying speed of light.MSC: 53C60, 83D05 83A05, 83C05.