Variationality of conformal geodesics in dimension 3
Differential Geometry
2026-04-07 v2 Mathematical Physics
Classical Analysis and ODEs
math.MP
Abstract
Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler--Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.
Keywords
Cite
@article{arxiv.2412.04890,
title = {Variationality of conformal geodesics in dimension 3},
author = {Boris Kruglikov and Vladimir S. Matveev and Wijnand Steneker},
journal= {arXiv preprint arXiv:2412.04890},
year = {2026}
}
Comments
A remark about conformally invariant Lagrangian is supplied at the end. More references added. Ancillary files can be accessed through version 1