On the geometry of curves and conformal geodesics in the Möbius space

Abstract

This article deals with the study of some properties of immersed curves in the conformal sphere Q n , viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in Q n lies, in fact, in a totally umbilical 4-sphere Q 4 . We then extend and complete the work in Musso (Math Nachr 165:107-131, 1994) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.

Quantization of the conformal arclength functional on space curves

“…Using Lemma 2.3, an explicit integration of the critical curves was carried out in [26] (see also [22]). The proof of Theorem 2.2 is a direct consequence of this integration.…”

“…In this paper we address the question of existence and properties of closed critical curves for the functional L. Actually, it suffices to consider the 3-dimensional case only, since from the results in [22] we can see that any closed critical curve in R n lies in some R 3 ⊂ R n , up to a conformal transformation. It is known that a generic space curve is determined, up to conformal transformations, by the conformal arclength and two conformal curvatures (cf.…”

The Cauchy–Riemann strain functional for Legendrian curves in the 3-sphere

“…Chains arise as projections of null geodesics of the Fefferman conformal structure. Inspired by the strong interrelationships between cr and Lorentzian conformal geometry and by some earlier works on conformal geometry of curves [13,36,38,43], we analyze global properties of Legendrian curves in the 3-sphere equipped with its standard cr-structure. In addition to the aforementioned interrelationships with Lorentzian conformal geometry, the fact that the cr-transformation group of S 3 is a real form of PSL(3, C), explains the many formal similarities with classical projective differential geometry of plane curves [6,26,42,40,46].…”

Conformal Geometry of Arc length Functional on Space Curves and Space time

“…We consider the conformally invariant variational problem on generic curves defined by the conformal arclength functional ℒ = . For = 3 and higher dimensions this variational problem was studied in [7], [11]. A generic space curve is determined, up to conformal transformations, by the conformal arclength and two conformal curvatures.…”