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

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.


53C22 Geodesics in global differential geometry
53B35 Local differential geometry of Hermitian and Kählerian structures
Full Text: DOI Euclid


[1] T. Adachi, Kähler magnetic flows for a manifold of constant holomorphic sectional curvature, Tokyo J. Math., 18 (1995), 473-483. · Zbl 0861.53070
[2] T. Adachi, A comparison theorem on magnetic Jacobi fields, Proc. Edinburgh Math. Soc. (2), 40 (1997), 293-308. · Zbl 0966.53047
[3] T. Adachi, Magnetic Jacobi fields for Kähler magnetic fields, In: Recent Progress in Differential Geometry and its Related Fields, Proceedings of the 2nd International Colloquium on Differential Geometry and its Related Fields, (eds. T. Adachi, H. Hashimoto and M. J. Hristov), World Scientific, 2011, pp.,41-53. · Zbl 1261.53068
[4] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, 9 , North-Holland Publ. Co., 1975. · Zbl 0309.53035
[5] N. Gouda, Magnetic flows of Anosov type, Tohoku Math. J. (2), 49 (1997), 165-183. · Zbl 0938.37011
[6] N. Gouda, The theorem of E. Hopf under uniform magnetic fields, J. Math. Soc. Japan, 50 (1998), 767-779. · Zbl 0914.53023
[7] K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163-170. · Zbl 0273.53039
[8] T. Sakai, Riemannian Geometry, Syokabo, 1992 (in Japanese) and Transl. Math. Monogr., 149 , Amer. Math. Soc., Providence, RI, 1996.
[9] T. Sunada, Magnetic flows on a Riemann surface, Proc. KAIST Math. Workshop (Analysis and geometry), 8 , 1993, pp.,93-108.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.