Curves with Constant Curvature Ratios

The notion of a generalized helix in R3, a curve making a constant angle with a fixed direction, can be generalized to higher dimensions in many ways. In [7] the same definition is proposed but in R. In [4] the definition is more restrictive: the fixed direction makes a constant angle with all the vectors of the Frenet frame. It is easy to check that this definition only works in the odd dimensional case. Moreover, in the same reference, it is proven that the definition is equivalent to the fact that the ratios k2 k1 , k4 k3 , . . . , ki being the curvatures, are constant. This statement is related with the Lancret Theorem for generalized helices in R3 (the ratio of torsion to curvature is constant). Finally, in [1] the author proposes a definition of a general helix in a 3-dimensional real-space-form substituting the fixed direction in the usual definition of generalized helix by a Killing vector field along the curve. In this paper we study the curves in R for which all the ratios k2 k1 , k3 k2 , k4 k3 , . . . are constant. We call them curves with constant curvature ratios or ccrcurves. The main result is that, in the even dimensional case, a curve has constant curvature ratios if and only if its tangent indicatrix is a geodesic in the flat torus. In the odd case, a constant must be added as the new coordinate function. In the last section we show that a ccr-curve in S3 is a general helix in the sense of [1] if and only if it has constant curvatures. To achieve this result, we have obtained the characterization of spherical curves in R4 in terms of the curvatures. Moreover, we have also found explicit examples of spherical ccr-curves with non-constant curvatures.