Affine projection tensor geometry: Lie derivatives and isometries

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.