A theorem of Hadamard-Cartan type for Kähler magnetic fields
Toshiaki ADACHI
Source: J. Math. Soc. Japan Volume 64, Number 3
(2012), 969-984.
Abstract
We study the global behavior of trajectories for Kähler magnetic fields on a connected complete Kähler manifold M of negative curvature. Concerning these trajectories we show that theorems of Hadamard-Cartan type and of Hopf-Rinow type hold: If 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.
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Keywords: Kähler magnetic fields; trajectories; trajectory-spheres; trajectory-harps; comparison theorem; theorem of Hopf-Rinow; theorem of Hadamard-Cartan
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1343133751
Digital Object Identifier: doi:10.2969/jmsj/06430969
Mathematical Reviews number (MathSciNet): MR2965435
References
T. Adachi, Kähler magnetic flows for a manifold of constant holomorphic sectional curvature, Tokyo J. Math., 18 (1995), 473–483.
Mathematical Reviews (MathSciNet): MR1363481
Digital Object Identifier: doi:10.3836/tjm/1270043477
Project Euclid: euclid.tjm/1270043477
T. Adachi, A comparison theorem on magnetic Jacobi fields, Proc. Edinburgh Math. Soc. (2), 40 (1997), 293–308.
Mathematical Reviews (MathSciNet): MR1454024
Zentralblatt MATH: 0966.53047
Digital Object Identifier: doi:10.1017/S0013091500023737
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.
Mathematical Reviews (MathSciNet): MR2882530
Digital Object Identifier: doi:10.1142/9789814355476_0003
J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, 9, North-Holland Publ. Co., 1975.
Mathematical Reviews (MathSciNet): MR458335
N. Gouda, Magnetic flows of Anosov type, Tohoku Math. J. (2), 49 (1997), 165–183.
Mathematical Reviews (MathSciNet): MR1447180
Digital Object Identifier: doi:10.2748/tmj/1178225145
Project Euclid: euclid.tmj/1178225145
Zentralblatt MATH: 0938.37011
N. Gouda, The theorem of E. Hopf under uniform magnetic fields, J. Math. Soc. Japan, 50 (1998), 767–779.
Mathematical Reviews (MathSciNet): MR1626370
Zentralblatt MATH: 0914.53023
Digital Object Identifier: doi:10.2969/jmsj/05030767
Project Euclid: euclid.jmsj/1225113732
K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163–170.
Mathematical Reviews (MathSciNet): MR348674
Zentralblatt MATH: 0273.53039
Digital Object Identifier: doi:10.1007/BF01360038
T. Sakai, Riemannian Geometry, Syokabo, 1992 (in Japanese) and Transl. Math. Monogr., 149, Amer. Math. Soc., Providence, RI, 1996.
Mathematical Reviews (MathSciNet): MR1390760
T. Sunada, Magnetic flows on a Riemann surface, Proc. KAIST Math. Workshop (Analysis and geometry), 8, 1993, pp.,93–108.
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