A generic smooth curve of the three dimensional Mobius space admits a natural parameter (or conformal arclength). Integrating the conformal arclength we get a conformally invariant variational problem. In the present paper we study the extremal curves of this variational problem. We derive the associated Euler-Lagrange equations and we get the natural equations of the extremal curves. The natural equations are integrated and the explicit solutions are given.
IntroductionIn 1980 C. SCHIEMANGK-R. SULANKE ([lo]) and in 1981 R. SULANKE ([ll]) considered the geometry of curves and surfaces in n-dimensional spheres with respect to the action of the group of all orientation-preserving conformal transformations.Section 3 of [lo] and all the paper [ll] are devoted to the geometry of curves in Mobius spaces (mainly in the 3-dimensional case). For generic curves a natural parametrization is founded, a natural lift into the corresponding conformal group is given (the conformal Frenet frame field of the curve) and generalized Frenet formulas are obtained.If we take a curve g(s) of lR3 parametrized by the euclidean arclength s, then the conformal parameter c is given by dc = (k" + k2z2)z ds, where k and z are the curvature and the torsion respectively and k' is the derivative of k with respect to the euclidean arclength s. If d{I, # 0 for all s then the curve is called generic.For generic curves we have that t; is a conformally invariant parametrization and we call the parameter ( the conformal arclength. There are two conformal curvatures, say k , and k , which together with t; give complete information of the geometry of the curve (up to conformal transformations of the ambient space).It seems that the conformal arclength and the conformal curvatures for curves in the 3-dimensional Mobius space were discovered earlier. In [lo] and [ll] the following references are given: H. LIEBMANN, 1923 [8], E. VESSIOT, 1925 [13] and T. TAKASU, 1928 [12].
The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations and the corresponding existence problem is discussed.2000 Mathematics Subject Classification. Primary 53A40, 53C24.
Abstract. We consider the variational problem defined by the functionaldA on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.
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