SLANT AND LEGENDRE CURVES IN BERGER su(2): THE LANCRET INVARIANT AND QUANTUM SPHERICAL CURVES

Abstract. Slant and Legendre curves are considered on Berger su(2) and are characterized through the scalar product between the normal at the curve and the vertical vector field; in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of these curves is obtained as well as their curvature and torsion. For the slant non-Legendre case we derive a Lancret-type invariant. By using the exponential map we obtain remarkable classes of curves on S3(1); in the helix case, and taking into account a B.-Y. Chen characterization of Legendre curves, we get a 1-parameter family of curves in relationship with the spectrum of the quantum harmonic oscillator. These curves, called by us quantum spherical curves, and their mates, provided by integer multiples of π, belong to antipodal Hopf fibres. 1