Length spectrum of circles in a complex projective space (Q1268677)

A circle with geodesic curvature \(\kappa\geq 0\) is a curve \(\gamma\) parametrized by arc-length in a Riemannian manifold and satisfying the equation \[ \nabla_t\nabla_t\dot\gamma(t)=-\kappa^2\dot\gamma(t). \] The authors restrict themselves to the complex projective space. They consider the set of prime periods of periodic circles and show that it coincides with the real positive line. If the geodesic curvature is kept fixed, then they show that this set is discrete and unbounded. If what the authors call complex torsion \(\tau\) is kept fixed and \(\kappa\) not, the set of prime periods is also discrete and unbounded. They calculate the smallest possible prime period if \(\kappa\) or \(\tau\) is kept fixed.