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.Comment: 15 page