Conformal geodesics on gravitational instantons

Maciej Dunajski,Paul Tod

doi:10.1017/s0305004121000463

Abstract

AbstractWe study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

Highlights

The geodesic flow on a Riemannian manifold (M, g) is integrable if the underlying metric admits a sufficient number of Killing vectors, or Killing tensors
In Appendix A we shall rule out the existence of non–flat Riemannian Gibbons–Hawking metrics with three commuting vector fields, and in Appendix B we shall provide necessary and sufficient conditions for a Killing trajectory to be a conformal geodesic
If the metric g is Einstein, i.e. R = (S/n)g the conformal geodesic equation (1·1) for a curve parametrised by an arc–lengh s, and with a unit tangent vector u reduces to

Summary

Introduction

The geodesic flow on a (pseudo) Riemannian manifold (M, g) is integrable if the underlying metric admits a sufficient number of Killing vectors, or Killing tensors. Where n = dim(M), and div = ∗d∗ is the divergence This was sufficient [25] to integrate (1·1) on a non–conformally flat Nil 3–manifold, and a squashed 3–sphere but already the conformal class of the Schwarzchild metric proved too difficult to handle, as there do not exist sufficiently many CKYs. In this paper we shall study the integrability of (1·1) on four–dimensional conformal structures corresponding to some gravitational instantons: solutions to Einstein equations on Riemannian four–manifolds which are compact or complete metrics which asymptotically approach a locally flat space - the decay rate, as well as the topology at infinity varries between different examples of instantons. In Appendix A we shall rule out the existence of non–flat Riemannian Gibbons–Hawking metrics with three commuting vector fields, and in Appendix B we shall provide necessary and sufficient conditions for a Killing trajectory to be a conformal geodesic

Conformal geodesics on Einstein manifolds

Circles on hyper–Kähler four–manifolds and the Lorentz force

Circles on anti–self–dual Taub–NUT

Note that the claim is not true in the neutral signature