  The generalized projection-tensor geometry introduced in an earlier paper is
extended. A compact notation for families of projected objects is introduced
and used to summarize the results of the previous paper and obtain fully
projected decompositions of Lie derivatives of the projection tensor field, the
metric and the projected parts of the metric. These results are applied to the
analysis of spacetimes with isometries. The familiar cases of spacetimes with
isotropic group orbits --- cosmological models and spherical symmetry --- are
discussed as illustrations of the results.
