Integrability of the geodesic flow on the resolved conifolds over Sasaki–Einstein space T1,1

Abstract

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds (rcs) over Sasaki–Einstein space [Formula: see text]. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki–Einstein space [Formula: see text] and its Calabi–Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution, the geodesic motion on the rc remains completely integrable. Instead, in the case of the deformation of the conifold, the complete integrability is lost.