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**Taub-NUT as Bertrand spacetime with magnetic fields.**
*(English)*
Zbl 1378.37097

This paper further elucidates the enigmatic Taub-NUT metric, a metric once described as “a counterexample to almost anything” (C. W. Misner, 1967). In particular, it studies Euclidean Taub-NUT metric and relates it, through its symmetries, to a the Euclidean Bertrand metric with magnetic monopoles and dipoles, and, in the process, it allows the understanding of both.

The paper’s outline is as follows. After a brief introduction, conserved quantities are considered using both a dynamical systems description of the Taub-NUT metric and a Holten algorithm description. Next Bertrand spacetimes with magnetic fields are described together with their Kepler-oscillator duality. The Taub-NUT metric is then derived from the self-dual Bianchi type-IX metric described by the Darboux-Halphen system. A discussion follows of curvature, anti-self-duality, and topological invariants. The Taub-NUT metric is interpreted as a gravitational instanton, allowing its study as an integrable system. An easy derivation and comparison between the spatial Killing-Yano tensors deduced from first integrals and the corresponding hyper-Kähler structure is given together with a verification of the existence of a graded Lie algebra structure via Schouten-Nijenhuis brackets.

Researchers in both general relativity and string theory will find this article enlightening.

The paper’s outline is as follows. After a brief introduction, conserved quantities are considered using both a dynamical systems description of the Taub-NUT metric and a Holten algorithm description. Next Bertrand spacetimes with magnetic fields are described together with their Kepler-oscillator duality. The Taub-NUT metric is then derived from the self-dual Bianchi type-IX metric described by the Darboux-Halphen system. A discussion follows of curvature, anti-self-duality, and topological invariants. The Taub-NUT metric is interpreted as a gravitational instanton, allowing its study as an integrable system. An easy derivation and comparison between the spatial Killing-Yano tensors deduced from first integrals and the corresponding hyper-Kähler structure is given together with a verification of the existence of a graded Lie algebra structure via Schouten-Nijenhuis brackets.

Researchers in both general relativity and string theory will find this article enlightening.

Reviewer: Deborah Konkowski (Annapolis)

### MSC:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

83C50 | Electromagnetic fields in general relativity and gravitational theory |

53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |

70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |

81R12 | Groups and algebras in quantum theory and relations with integrable systems |