Slicing, Threading & Parametric Manifolds
  We present a unified treatment of the slicing (3+1) and threading (1+3) decompositions of spacetime in terms of foliations. It is well-known how to decompose the metric and connection in the slicing picture; this is at the heart of any initial-value problem in general relativity. We describe here the analogous problem in the threading picture, recovering the recent results of Perjes on parametric manifolds. 
Stuart Boersma and Tevian Dray