A parallel null direction in higher-order symmetric spacetimes
A Lorentzian manifold with Riemann curvature tensor $R$ is properly $r$th-order symmetric if $\nabla^r R = 0$ and $\nabla^{r-1} R \not\equiv 0$, where $\nabla^j R$ is the full iterated covariant derivative, with no symmetrization. Senovilla proved that a simply connected domain of a second-order symmetric Lorentzian manifold is locally symmetric or carries a parallel null vector field (2008), and Galaev proved that every simply connected properly third-order symmetric Lorentzian manifold admits a parallel null vector field (2015). For every $r$, plane waves whose profile is a polynomial of degree $r - 1$ in the null coordinate are $r$th-order symmetric and have a parallel null vector field (Senovilla 2008, 2024).
Problem. Let $r \ge 4$ and $n \ge 3$ be integers, and let $(M^n, g)$ be a connected simply connected smooth Lorentzian manifold which is properly $r$th-order symmetric. Must there exist a smooth vector field $V$ on $M$ with $V_p \ne 0$ for all $p \in M$, $g(V, V) = 0$ and $\nabla V = 0$?
Source: Senovilla (2008), Conjecture 1, for every order on simply connected domains; see also Senovilla (2024), Conjecture 2, and Galaev (2015). Stated by AIM — Open Applied Problems for proper order at least four on a simply connected manifold.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, higher-order symmetric spaces, Lorentzian manifolds, parallel null vector fields
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{senovilla2008,
author = {Senovilla, Jos{\'e} M. M.},
title = {Second-order symmetric {L}orentzian manifolds: {I}. {C}haracterization and general results},
journal = {Classical Quantum Gravity},
volume = {25},
number = {24},
year = {2008},
pages = {245011},
doi = {10.1088/0264-9381/25/24/245011},
eprint = {math/0604113},
archivePrefix = {arXiv},
}
@article{galaev2015,
author = {Galaev, Anton S.},
title = {Classification of third-order symmetric {L}orentzian manifolds},
journal = {Classical Quantum Gravity},
volume = {32},
number = {2},
year = {2015},
pages = {025001},
doi = {10.1088/0264-9381/32/2/025001},
eprint = {1407.3594},
archivePrefix = {arXiv},
}
@article{senovilla2024,
author = {Senovilla, Jos{\'e} M. M.},
title = {Semi-{R}iemannian manifolds with linear differential conditions on the curvature},
journal = {Anal. Math. Phys.},
volume = {14},
number = {3},
year = {2024},
pages = {63},
doi = {10.1007/s13324-024-00923-0},
eprint = {2308.15436},
archivePrefix = {arXiv},
}
@misc{aim-a-parallel-null-direction-in-higher-order-symmetric-spacetimes,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A parallel null direction in higher-order symmetric spacetimes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/874-a-parallel-null-direction-in-higher-order-symmetric-spacetimes.md},
note = {Entry 874 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01109, "A parallel null direction in higher-order symmetric spacetimes", clopen.solutions.