Comparison on bow-shapes for Kaehler magnetic fields

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The head of investigator mainly studied real 2-dimensional surfaces which are naturally obtained from trajectories for Kaehler magnetic fields. He compared such surfaces on complex space forms with such surfaces on general Kaehler manifolds..(1)Comparison on bow-shapes and crescentsGiven a trajectory for a Kaehler magnetic field, we attach at each point on this trajectory a geodesic whose initial vector and the tangent vector of trajectory span a complex line in the tangent space. Such geodesics form a real 2-dimensional surfece. On a complex space form this surface is a complex space line. In order to study tie shape of this surfece on a general Kaehler manifold, for a part of this trajectory we take a geodesic segment on this surfece which joins its ends. Under the condition of sectional curvature from above, we find the length of the geodesic segment is not shorter than the length of bow-shape on a complex space form. We also find that the equality holds if and only if the crescent … More is totally geodesic immersed bow-shape.(2)Comparison on sectorsWe consider an image of a complex line through magnetic exponential map. On a complex space form the image of r-ball in a tangent complex line is an intersection of geodesic r-ball and a complex space line. In order to study this sector on a general Kaehler manifold, we consider the length of arc of this sector. Under the condition of sectional curvature from below, we find the length of the arc is not longer than the length of arc of a corresponding sector on a complex space form. We also find that the equality holds if and only if the sector is totally geodesic and complex embedded one.(3)Variation of force and pseudo-congruency of Kaehler magnetic flowsThough we have a real 2-dimensional surface which is obtained by varying forces of Kaehler magnetic fields, it is not so natural on a general Kaehler manifold in view of mappings. Using this property we studied pseudo-congruency of Kaehler magnetic flows on a Kaehler manifold with complex Euclidean factor. As an application we characterize the rank of Hermitian symmetric space of noncompact type in terms of force of Kaehler magnetic field with horocycle trajectories. Less