On Geometry and Symmetry of Kepler Systems. I

Abstract

We study the Kepler metrics on Kepler manifolds from the point of view of Sasakian geometry and Hessian geometry. This establishes a link between the problem of classical gravity and the modern geometric methods in the study of AdS/CFT correspondence in string theory.

Noncommutative Kepler Dynamics: symmetry groups and bi-Hamiltonian structures

“…In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the solar system by relating the five extra-terrestrial planets (Mercury, Venus, Mars, Jupiter, and Saturn) known at that time to be the five Platonic solids (the tetrahedron or pyramid, cube, octahedron, dodecahedron, and icosahedron) [40,61]. Kepler's work attracted the attention of the Danish astronomer Brahe who recruited him in October 1600 as an assistant.…”

Hessian Geometry and Phase Change of Gibbons-Hawking Metrics

“…This is a sequel to [20] in which techniques developed in string theory were applied to study the Kepler problem in classical gravity. We will apply the same techniques in this paper to study gravitational instantons in Euclidean gravity in dimension four of type A n−1 , i.e., the Gibbons-Hawking metrics.…”

“…This leads to the applications of Hessian geometry [16] on convex cones to AdS/CFT [15]. These techniques were generalized and applied to the Kepler problem in [20]. First, in §6 of that work, we use the explicit construction of Kähler metrics with U (n)symmetry [12,5,6] to obtain applications of symplectic coordinates and Hessian geometry.…”

“…An alternative treatment, based on Calabi ansatz [2], is presented in [20, §8]. Generalizations to the Kepler metric and the Kähler Ricci-flat metrics on the resolved conifold are made in §9 and §10 of [20], based on Calabi ansatz on O P 1 (−1) ⊕ O P 1 (−1) and O P 1 ×P 1 (−1, −1) respectively. The basic observations in [20] is that all these metrics obtained by such explicit constructions are toric and the images of their moment maps are polyhedral convex bodies, and one can recover the complex structures and Kähler structures by Hessian structures [16] on the convex bodies.…”

“…In [20, §10.6], the author presented a new method to describe the flop of Kähler Ricciflat metrics on the resolved conifold [3] by embedding them in a two-parameter family of Kähler Ricci-flat metrics on the canonical line bundle of P 1 × P 1 . It then becomes natural to reexamine the phase change phenomena introduced in [5,6] from the point of view of [20]. This has been carried out recently for local P 1 , local P 2 and local P 1 × P 1 cases in [18].…”