Conformal Curvatures
by
Esther Sanabria-Codesal
Universidad de Valencia, Spain
Coauthors: Angel Montesinos Amilibia (Universidad de Valencia), M Carmen Romero Fuster (Universidad de Valencia)

Poster

R. Sulanke proved in [8] the fundamental theorem for generic curves in the Möbius space by using the Cartan's method of moving frames. It follows from this the existence of a system of conformal invariants for curves in (n+1)-space: the conformal curvatures. A. Fialkow also defined in [4] a set of conformal curvatures by using conformal derivations techniques.

It is quite natural to ask for relations between these and the classical Euclidean invariants. In this sense, G. Cairns, R. W. Sharpe and L. Webb display in [2] an expression for the conformal torsion of a curves in 3-space in terms of its Euclidean curvature and torsion. The generalization of their method for n > 3, leads to very complicated and somehow unhandlable expressions. Nevertheless we observed in [7] that these procedure could be surprisingly simplified if one works with the centers and the radii of the osculating hyperspheres instead of the Euclidean curvatures. We actually obtained the expression for what we called the conformal torsion (or nth conformal curvature) of a curve in (n+1)-space that generalizes the one . This was done by using Coxeter's inversive distance between pairs of n-spheres in IRⁿ⁺¹.

We develop in [6] a theory of conformal invariants for pairs of spheres (with not neccessarily the same dimensions) in (n+1)-space. These play the role of the inversive distance between hyperspheres (see [3] and [1]) and by applying them to the families of osculating k-spheres (1 <= k <= n) of generic curves in (n+1)-space, we obtain n conformally invariant 1-forms along the curve. These in turn give us some functions (rational expressions of the radii of the Euclidean curvatures) which can be interpreted as the conformal arclength if k = 1 on the one hand, and the kth conformal curvatures, 2 <= k <= n, on the other in the sense that they also measure how far the curve is from lying on a k-sphere. R. Sulanke has also found a system of conformal invariants for pairs of spheres in (n+1)-space by using different methods ([9]).

We notice that the first conformal curvature cannot be described by this method. A geometrical interpretation of this function has been given by Sulanke in [8], whereas its expression in terms of the Euclidean curvatures in a more general setting (conformal maps between general Riemmanian spaces) was already displayed in [4]. It is also worth mentioning that the conformal arclength was already known as a function of the first Euclidean curvature (see [4] and [8]). In fact, it first appeared in the paper [5] of Liebmann in 1923.

It is not evident at all that our functions must coincide with those formally defined by Sulanke or Fialkow's methods. Nevertheless, it follows from our arguments that they must vanish on the same set of points (conformal vertices).

Bibliography

1 A. F. Beardon. The geometry of discrete groups. Springer Verlag. (1983).
2 G. Cairns, R. W. Sharpe and L. Webb. Conformal invariants for curves and surfaces in three dimensional space forms. Rocky Mountain Journal of Math. 24 (1994), 933-959.
3 H. S. M. Coxeter. Inversive distance. Ann. Mat. Pura Apli. 4, 71 (1966), 73-83.
4 A. Fialkow. The conformal theory of curves. Trans. Amer. Math. Soc. 51 (1942), 435-501.
5 H. Liebmann. Beiträge zur Inversionsgeometrie der Kurven. Muenchener berichte. (1923) 79-94.
6 A. Montesinos Amilibia, M. C. Romero Fuster and E. Sanabria Codesal, Conformal curvatures of curves in IRⁿ⁺¹. Preprint.
7 M. C. Romero-Fuster and E. Sanabria-Codesal. Generalized evolutes, vertices and conformal invariants of curves in IRⁿ⁺¹. Indag. Mathem. N.S. 10, 2 (1999), 297-305.
8 R. Sulanke. Submanifolds of the Möbius space II, Frenet Formulas and Curves of Constant Curvatures. Math. Nachr. 100 (1981), 235-247
9 R. Sulanke. Möbius invariants for pairs (S₁^m, S₂^l) of spheres in the Möbius space Sⁿ. To appear in Beiträge zur Algebra und Geometrie.

Date received: May 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-92.