Accurate trajectory-harps for Kähler magnetic fields. (English) Zbl 1405.53057

Summary: In preceding papers we gave estimates on string-lengths, string-cosines and zenith angles of trajectory-harps under the condition that sectional curvatures of the underlying manifold are bounded from above. In this paper we study the cases that equalities hold in these estimates. Refining the previous proofs we give conditions that trajectory-harps are congruent to trajectory-harps on a complex space form.


53C22 Geodesics in global differential geometry
53B35 Local differential geometry of Hermitian and Kählerian structures
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[1] T. Adachi, Kähler magnetic flows on a manifold of constant holomorphic sectional curvature, Tokyo J. Math., 18 (1995), 473–483. · Zbl 0861.53070
[2] T. Adachi, A comparison theorem for magnetic Jacobi fields, Proc. Edinburgh Math. Soc., 40 (1997), 293–308. · Zbl 0966.53047
[3] T. Adachi, A theorem of Hadamard–Cartan type for Kähler magnetic fields, J. Math. Soc. Japan, 64 (2012), 969–984. · Zbl 1252.53047
[4] T. Adachi, A comparison theorem on harp-sectors for Kähler magnetic fields, Southeast Asian Bull. Math., 38 (2014), 619–626. · Zbl 1324.53018
[5] T. Adachi and P. Bai, Volumes of trajectory-balls for Kähler magnetic fields, J. Geom., 105 (2014), 369–389. · Zbl 1302.53044
[6] P. Bai and T. Adachi, An estimate of the spread of trajectories for Kähler magnetic fields, Hokkaido Math. J., 42 (2013), 445–462. · Zbl 1281.53041
[7] A. Comtet, On the Landau levels on the hyperbolic plane, Ann. Phys., 173 (1987), 185–209. · Zbl 0635.58034
[8] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, Noth-Holland, 1975, Amsterdam. · Zbl 0309.53035
[9] T. Sunada, Magnetic flows on a Riemann surface, Proc. KAIST Math. Workshop, 8 (1993), 93–108.
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