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 metrics 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.
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Derdzinski, A., Masood, K. Null-Projectability of Levi-Civita Connections. Results Math 78, 141 (2023). https://doi.org/10.1007/s00025-023-01911-9
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DOI: https://doi.org/10.1007/s00025-023-01911-9