Conformal geodesics on gravitational instantons

@article{Dunajski2019ConformalGO,
  title={Conformal geodesics on gravitational instantons},
  author={Maciej Dunajski and Paul Tod},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2019},
  volume={173},
  pages={123 - 154}
}
  • M. DunajskiP. Tod
  • Published 19 June 2019
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract We 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… 

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References

SHOWING 1-10 OF 28 REFERENCES

Self-Dual Conformal Gravity

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M, g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci-flat manifold. These conditions are

Conformal circles and parametrizations of curves in conformal manifolds

We give a simple ODE for the conformal circles on a conformal manifold, which gives the curves together with a family of preferred parametrizations. These parametrizations endow each conformal circle

Black holes, hidden symmetries, and complete integrability

It is demonstrated that the principal tensor can be used as a “seed object” which generates all these symmetries of higher-dimensional Kerr–NUT–(A)dS black hole spacetimes and the review contains a discussion of different applications of the developed formalism and its possible generalizations.

Kähler Magnetic Flows for a Manifold of Constant Holomorphic Sectional Curvature

In his paper [7], being inspired by a classical treatment of static magnetic fields in the three dimensional Euclidean space, T. Sunada studied the flow associated with a magnetic field on a Riemann

Variational principles for conformal geodesics

Conformal geodesics are solutions to a system of third-order equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational

Integrability Versus Separability for the Multi-Centre Metrics

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra

Conformal theory of curves with tractors

Solitons, Instantons, and Twistors

Preface 1. Integrability in classical mechanics 2. Soliton equations and the Inverse Scattering Transform 3. The hamiltonian formalism and the zero-curvature representation 4. Lie symmetries and

On the Landau Levels on the Hyperbolic Plane

Circles in a complex projective space

The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field