This paper studies the geometry of the critical points of the simplest conformally invariant variational problem for timelike curves in the n-dimensional Einstein universe. Such critical curves are referred to as conformal timelike geodesics. The functional defining the variational problem is the Lorentz analogue of the conformal arclength functional in M"obius geometry. We compute the Euler--Lagrange equations and show that the trajectory of a conformal timelike geodesic is constrained into some totally umbilical Einstein universe of dimension 2, 3, or 4. The case of dimension 2 leads to orbits of 1-parameter groups of Lorentz Moebius transformations, while that of dimension 3 has been dealt with in [Nonlinear Anal. 143 (2016), 224-255]. In this paper, we discuss the case of conformal timelike geodesics in the 4-dimensional Einstein universe whose trajectories are not contained in any lower dimensional totally umbilical Einstein universe. It is shown that such curves can be explicitly integrated by quadratures and explicit expressions in terms of elliptic functions and integrals are provided.

The geometry of conformal timelike geodesics in the Einstein universe / Eshkobilov, O.; Musso, E.; Nicolodi, L.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 495:2(2021), pp. 1-32. [10.1016/j.jmaa.2020.124730]

The geometry of conformal timelike geodesics in the Einstein universe

L. Nicolodi
2021

Abstract

This paper studies the geometry of the critical points of the simplest conformally invariant variational problem for timelike curves in the n-dimensional Einstein universe. Such critical curves are referred to as conformal timelike geodesics. The functional defining the variational problem is the Lorentz analogue of the conformal arclength functional in M"obius geometry. We compute the Euler--Lagrange equations and show that the trajectory of a conformal timelike geodesic is constrained into some totally umbilical Einstein universe of dimension 2, 3, or 4. The case of dimension 2 leads to orbits of 1-parameter groups of Lorentz Moebius transformations, while that of dimension 3 has been dealt with in [Nonlinear Anal. 143 (2016), 224-255]. In this paper, we discuss the case of conformal timelike geodesics in the 4-dimensional Einstein universe whose trajectories are not contained in any lower dimensional totally umbilical Einstein universe. It is shown that such curves can be explicitly integrated by quadratures and explicit expressions in terms of elliptic functions and integrals are provided.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2872682
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact