  We consider Riemannian $n$-manifolds $M$ with nontrivial $\kappa$-nullity
"distribution" of the curvature tensor $R$, namely, the variable rank
distribution of tangent subspaces to $M$ where $R$ coincides with the curvature
tensor of a space of constant curvature $\kappa$ ($\kappa\in\mathbb R$) is
nontrivial. We obtain classification theorems under diferent additional
assumptions, in terms of low nullity/conullity, controlled scalar curvature or
existence of quotients of finite volume. We prove new results, but also revisit
previous ones.
