A theorem of Hadamard-Cartan type for Kähler magnetic fields. (English) Zbl 1252.53047

Adachi, Toshiaki

J. Math. Soc. Japan 64, No. 3, 969-984 (2012).

The author studies the global behavior of trajectories for Kähler magnetic fields on a connected complete Kähler manifold \(M\) of negative curvature. He shows that theorems of Hadamard-Cartan type and of Hopf-Rinow type hold for such trajectories. More precisely, if the sectional curvatures of \(M\) are not greater than \(c\) (\(< 0\)) and the strength of a Kähler magnetic field is not greater than \(\sqrt{|c|}\), then every magnetic exponential map is a covering map. Hence arbitrary distinct points on \(M\) can be joined by a minimizing trajectory for this magnetic field.

Reviewer: A. Arvanitoyeorgos (Patras)

MSC:

53C22	Geodesics in global differential geometry
53B35	Local differential geometry of Hermitian and Kählerian structures

Keywords:

Kähler magnetic fields; trajectories; trajectory-spheres; trajectory-harps; comparison theorem; theorem of Hopf-Rinow; theorem of Hadamard-Cartan

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