Null-projectability of Levi-Civita connections

TL;DR

This paper investigates when Levi-Civita connections of pseudo-Riemannian metrics are projectable along null parallel distributions, extending classical results to broader cases with a generalized Riemann extension framework.

Contribution

It generalizes the characterization of projectable Levi-Civita connections to null distributions of any dimension, introducing a new class of Riemann extension metrics.

Findings

Characterization of projectability for null distributions of any dimension

Extension of Riemann extension metrics to broader null distributions

Connection between projectability and generalized Riemann extensions

Abstract

We study the natural property of projectability of a torsion-free connection along a foliation on the underlying manifold, which leads to a projected torsion-free connection on a local leaf space, focusing on projectability of Levi-Civita connections of pseudo-Riemannian metric along foliations tangent to null parallel distributions. For the neutral metric signature and mid-dimensional distributions, Afifi showed in 1954 that projectability of the Levi-Civita connection characterizes, locally, the case of Patterson and Walker's Riemann extension metrics. We extend this correspondence to null parallel distributions of any dimension, introducing a suitable generalization of Riemann extensions.