The space-time foliation compatible with the gravitational eld g on a 4-manifold M determines a bration of M , : M ! N is surjective submersion over the 1-dimensional leaves space N . M is then written as a disjoint union of the leaves of , which are 3-dimensional spacelike surfaces on M . The decomposition, TM = T 0 M , also implies that we can dene a lift of the curves on N to curves (non-spacelike) on M . The stable causality condition holds on M  means that is a causal space-time distribution, generated by an exact timelike 1-form ! 0 = dt where t is some real function on M . In this case M is written as a disjoint union of a family of spacelike 3-surfaces of constant t, which cover D + (S) of a initial 3-surface S of M . A.M.S. Classication Code. Primary: 83C99, Secondary: 53B30 53C50 53C12 57R30 Keywords. space-time distribution, 4-manifold Suceava, 5800 Romania E-mail: firstname.lastname@example.org Copyright c 1999 by Cameron University 23 1 Introduction Note:..