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

TL;DR

This paper classifies Riemannian manifolds with a nontrivial $ppa$-nullity distribution of the curvature tensor, providing new results and revisiting prior classifications under various geometric conditions.

Contribution

It introduces new classification theorems for manifolds with $ppa$-nullity, extending previous work and considering additional geometric assumptions.

Findings

Classification theorems under various nullity and curvature conditions

New results on manifolds with specific nullity properties

Revisiting and extending previous classification results

Abstract

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.