  It is well known that every Killing vector is a Ricci and Matter
collineation. Therefore the metric, the Ricci tensor and the energy-momentum
tensor are all members of a large family of second order symmetric tensors
which are invariant under a common group of symmetries. This family is
described by a generic metric which is defined from the symmetry group of the
space-time metric. The proper Ricci and Matter (inheritance) collineations are
the (conformal) Killing vectors of the generic metric which are not (conformal)
Killing vectors of the space-time metric. Using this observation we compute the
Ricci and Matter inheritance collineations of the Robertson-Walker space-times
and we determine the Ricci and Matter collineations without any further
calculations. It is shown that these higher order symmetries can be used as
supplementary conditions to produce an equation of state which is compatible
with the Geometry and the Physics of the Robertson-Walker space-times.
