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 $S^3(1)$; 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 $\pi$, belong to antipodal Hopf fibres.