Inheriting geodesic flows

We investigate the propagation equations' for the expansion, vorticity and shear for perfect fluid space-times which are geodesic. It is assumed that space-time admits a conformal Killing vector which is inheriting so that fl uid flow lines are mapped conformally. Simple constraints on the electric a nd magnetic parts of the Weyl tenser are found for conformal symmetry. For homothetic vectors the vorticity and shear are free; they vanish for nonhom othetic vectors. We prove a conjecture for conformal symmetries in the spec ial case of inheriting geodesic flows: there exist no proper conformal Kill ing vectors (psi (;ab) not equal 0) for perfect fluids except for Robertson -Walker space-times. For a nonhomothetic vector field the propagation of th e quantity ln(R(ab)u(a)u(b)) along the integral curves of the symmetry vect or is homogeneous.