The \(\kappa \)-nullity of Riemannian manifolds and their splitting tensors

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.