Completeness under conformal changes of compact Lorentzian spaceforms
Let $M$ be a closed connected smooth manifold of dimension $n \ge 2$ and $g_0$ a smooth Lorentzian metric on $M$ of constant sectional curvature; for an arbitrary $u \in C^\infty(M; \mathbb{R})$ set $g = e^{2u} g_0$.
The metric $g_0$ is geodesically complete, by Carrière in the flat case and by Klingler in general. On a compact manifold lightlike completeness is a conformal invariant, so the lightlike geodesics of $g$ are complete. Romero and Sánchez proved that a compact Lorentzian manifold is complete if it is globally conformal to a homogeneous one or admits a timelike conformal vector field.
Problem. Must $(M, g)$ be geodesically complete, that is, for every $x \in M$ and $v \in T_x M$, must the maximal solution of $\nabla^g_{\dot\gamma} \dot\gamma = 0$, $\gamma(0) = x$, $\dot\gamma(0) = v$ be defined for all $t \in \mathbb{R}$?
Source: Candela and Sánchez (2008), Introduction, for all semi-Riemannian signatures; in Lorentzian form Sánchez (2015), §4.4, Question 1, repeated as Question 9.3.2 by Burns and Matveev (2021).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, conformal changes, geodesic completeness, Lorentzian spaceforms
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{candela2008,
author = {Candela, Anna Maria and S{\'a}nchez, Miguel},
title = {Geodesics in semi-{R}iemannian manifolds: geometric properties and variational tools},
booktitle = {Recent Developments in Pseudo-{R}iemannian Geometry},
editor = {Alekseevsky, Dmitri V. and Baum, Helga},
series = {ESI Lectures in Mathematics and Physics},
publisher = {European Mathematical Society},
year = {2008},
pages = {359--418},
doi = {10.4171/051-1/10},
eprint = {math/0610144},
archivePrefix = {arXiv},
}
@incollection{sanchez2015,
author = {S{\'a}nchez, Miguel},
title = {On the completeness of trajectories for some mechanical systems},
booktitle = {Geometry, Mechanics, and Dynamics},
series = {Fields Institute Communications},
volume = {73},
publisher = {Springer},
year = {2015},
pages = {343--372},
doi = {10.1007/978-1-4939-2441-7_15},
eprint = {1303.6150},
archivePrefix = {arXiv},
}
@article{burns2021,
author = {Burns, Keith and Matveev, Vladimir S.},
title = {Open problems and questions about geodesics},
journal = {Ergodic Theory Dynam. Systems},
volume = {41},
number = {3},
year = {2021},
pages = {641--684},
doi = {10.1017/etds.2019.73},
eprint = {1308.5417},
archivePrefix = {arXiv},
}
@article{carriere1989,
author = {Carri{\`e}re, Yves},
title = {Autour de la conjecture de {L}. {M}arkus sur les vari{\'e}t{\'e}s affines},
journal = {Invent. Math.},
volume = {95},
number = {3},
year = {1989},
pages = {615--628},
doi = {10.1007/BF01393894},
}
@article{klingler1996,
author = {Klingler, Bruno},
title = {Compl{\'e}tude des vari{\'e}t{\'e}s lorentziennes {\`a} courbure constante},
journal = {Math. Ann.},
volume = {306},
number = {1},
year = {1996},
pages = {353--370},
doi = {10.1007/BF01445255},
}
@article{romero1994,
author = {Romero, Alfonso and S{\'a}nchez, Miguel},
title = {On completeness of certain families of semi-{R}iemannian manifolds},
journal = {Geom. Dedicata},
volume = {53},
number = {1},
year = {1994},
pages = {103--117},
doi = {10.1007/BF01264047},
}
@article{romero1995,
author = {Romero, Alfonso and S{\'a}nchez, Miguel},
title = {Completeness of compact {L}orentz manifolds admitting a timelike conformal {K}illing vector field},
journal = {Proc. Amer. Math. Soc.},
volume = {123},
number = {9},
year = {1995},
pages = {2831--2833},
doi = {10.1090/S0002-9939-1995-1257122-3},
}
@misc{aim-completeness-under-conformal-changes-of-compact-lorentzian-spaceforms,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Completeness under conformal changes of compact Lorentzian spaceforms}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/884-completeness-under-conformal-changes-of-compact-lorentzian-spaceforms.md},
note = {Entry 884 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-01112, "Completeness under conformal changes of compact Lorentzian spaceforms", clopen.solutions.