Conformal geodesics are not variational in higher dimensions
Abstract
Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler-Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.
Cite
@article{arxiv.2505.06739,
title = {Conformal geodesics are not variational in higher dimensions},
author = {Boris Kruglikov},
journal= {arXiv preprint arXiv:2505.06739},
year = {2025}
}
Comments
This version corrects some inconsistencies in notations, we also changed notations whenever a confusion between parameters and quantities could be possible. A historical remark on nomenclature was added at the end of the paper, and the list of references was updated