---
abstract: 'We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-dimensional Einstein universe and we show that they can be integrated by quadratures in terms of elliptic functions, elliptic integrals and Jacobi’s theta functions.'
address:
- '(O. Eshkobilov) Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10 - I-10123 Torino, Italy'
- '(E. Musso) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy'
author:
- Olimjon Eshkobilov
- Emilio Musso
date: Version of 
title: 'A conformally invariant variational problem for time-like curves'
---

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[^1]

Introduction {#s:intro}
============

Lorentzian conformal geometry began in $1918$ with the seminal work of H.Weyl [@W]. In the 1980s, it has been considerably developed in connection with the twistor approach to gravity by Penrose and Rindler [@PR] and it is widely used in the cyclic cosmological models in general relativity [@Pn1; @Pn3; @Tod]. It is also related to regularization of the Kepler problem [@GS] and Lie sphere geometry [@Blaschke; 