  A totally umbilical submanifold in pseudo-Riemannian manifolds is a
fundamental notion, which is characterized by the condition that the second
fundamental form is proportional to the metric. It is also a generalization of
the notion of a totally geodesic submanifold. In this paper, we classify
congruent classes of full totally umbilical submanifolds in non-flat
pseudo-Riemannian space forms, and consider their moduli spaces. As a
consequence, we show that some moduli spaces of isometric immersions between
space forms which one of the same constant curvature are non-Hausdorff.
