Conformal Geodesics Cannot Spiral
Differential Geometry
2025-07-25 v3 General Relativity and Quantum Cosmology
High Energy Physics - Theory
Mathematical Physics
math.MP
Abstract
We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.
Keywords
Cite
@article{arxiv.2205.07978,
title = {Conformal Geodesics Cannot Spiral},
author = {Peter Cameron and Maciej Dunajski and Paul Tod},
journal= {arXiv preprint arXiv:2205.07978},
year = {2025}
}
Comments
Wojciech Kami\'nski has provided a non real--analytic counterexample to Theorem 2.3. Apart from the missing assumption of real--analyticity the error is contained in Lemma 4.6 where it was claimed that the size of the heart is a continuous function. Other results the paper, in particular the definition of the exponential map, are unaffected by the above error.