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

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.

14 pages

References in corpus (2)

• Sasaki-Einstein Manifolds and Volume Minimisation
• Massless geodesics in $Ad{S}_5\times Y(p,q)$ as a superintegrable system

Cited by in corpus (4)

• Integrability of geodesics and action-angle variables in Sasaki-Einstein space $T^{1,1}$
• Sasaki-Ricci flow equation on five-dimensional Sasaki-Einstein space $Y^{p,q}$
• Action-angle variables for geodesic motions in Sasaki-Einstein spaces $Y^{p,q}$
• Integrability of the geodesic flow on the resolved conifolds over Sasaki-Einstein space $T^{1,1}$