Abstract
In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow. We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q.
Highlights
The paper is organized as follows: we start by recalling some background in Sasaki geometry, deformations of Sasaki metrics, and Sasaki–Ricci flow
We introduce a local set of transverse complex coordinates appropriate for the transverse Kähler structure of Y p,q [24,25]
Having in mind that Sasaki–Einstein manifolds have become of significant interest in many areas of physics, we investigate the integrability in the frame of contact geometry
Summary
Over the past four decades, contact geometry has undergone a rapid development in pure mathematics [1] and in applied areas as mechanics, dissipative systems, optics, thermodynamics, or control theory [2]. As it is well-known, the description of Hamiltonian mechanics is developed on symplectic manifolds. The paper is organized as follows: we start by recalling some background in Sasaki geometry, deformations of Sasaki metrics, and Sasaki–Ricci flow.
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