Complete integrability of geodesic motion in Sasaki-Einstein toric $Y^{p,q}$ spaces

TL;DR

This paper explicitly constructs constants of motion for geodesics in 5D Sasaki-Einstein $Y^{p,q}$ spaces, demonstrating their complete integrability through Killing vectors and tensors, with detailed analysis for the special case of $T^{1,1}$.

Contribution

It provides the first explicit construction of constants of motion for geodesics in $Y^{p,q}$ spaces, proving their complete integrability using Killing structures.

Findings

Geodesic flow in $Y^{p,q}$ spaces is completely integrable.

Only five constants of motion are functionally independent.

Simplified integrals of motion are obtained for $T^{1,1}$.

Abstract

We construct explicitly the constants of motion for geodesics in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. To carry out this task we use the knowledge of the complete set of Killing vectors and Killing-Yano tensors on these spaces. In spite of the fact that we generate a multitude of constants of motion, only five of them are functionally independent implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces. In the particular case of the homogeneous Sasaki-Einstein manifold $T^{1,1}$ the integrals of motion have simpler forms and the relations between them are described in detail.