Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data

Working within the ADM formalism we have established results characterizing a conformal Killing vector (CKV) field on a spacetime solution of Einstein's field equations and the initial data on a spacelike hypersurface. We have considered the cases (i) the CKV is tangential to spacelike slices, (ii) the CKV is closed and the initial hypersurface is totally umbilical and has constant mean curvature (CMC), and (iii) spacetime is a perfect fluid, has a non-vanishing CKV transverse to a compact CMC hypersurface orthogonal to the 4-velocity. Finally, we show that a non-null CKV that is also a geodesic vector field is either homothetic or locally closed.