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

ISSN： 0232-704X

Source： Annals of Global Analysis and Geometry, Vol.40, Iss.2, 2011-08, pp. : 133-165

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Abstract

This article deals with the study of some properties of immersed curves in the conformal sphere $${mathbb{Q}_n}$$ , 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 $${mathbb{Q}_n}$$ lies, in fact, in a totally umbilical 4-sphere $${mathbb{Q}_4}$$ . 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.