  We study the varieties of invariant totally geodesic submanifolds of
isometries of the spherical, Euclidean and hyperbolic spaces in each finite
dimension. We show that the dimensions of the connected components of these
varieties determine the orbit type (or the z-class) of the isometry. For this
purpose, we introduce the Segre symbol of an isometry, a discrete invariant
encoding the structure of its normal form, which parametrizes z-classes. We
then provide a description of the isomorphism type of the varieties of
invariant subspaces in terms of the Segre symbol.
