Complete Integrability of Geodesics in Sasaki–Einstein Space T1,1 and Its Resolved Conifold

Abstract

In this paper we are concernedwith integrability of geodesic motions in Sasaki–Einstein space T^1,1 and its Calabi–Yau metric cone. There are enough functionally independent integrals of motions to ensure the complete integrability for geodesics in T^1,1 space and its metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. In the case of the small resolution the geodesic motions on the resolved conifold remain completely integrable. However, in the case of the deformation of the conifold the complete integrability is lost.