Comparison on bow-shapes for Kaehler magnetic fields

研究负责人主要研究了实际的二维表面，这些表面是由卡勒磁场的轨迹自然得到的。他比较了复杂空间形式上的曲面和一般Kaehler流形上的曲面。(1)弓形和新月形的比较给定一个Kaehler磁场的轨迹，我们在该轨迹的每一点上附加一个测地线，测地线的初始矢量和轨迹的切矢量在切空间中跨越一条复线。这样的测地线形成了一个真正的二维曲面。在复空间形式中，这个曲面是一条复空间线。为了研究一般Kaehler流形上曲面的形状，对于这个轨迹的一部分，我们在这个曲面上取一个连接其两端的测地线段。由上可知，在截面曲率条件下，测地线线段的长度不小于复空间形式上的弓形长度。我们还发现，当且仅当新月是完全测地线浸没弓形时，等式成立。(2)扇区比较我们通过磁指数图考虑一条复线的图像。在复空间形式上，r球在切复直线上的像是测地线r球与复空间直线的交点。为了研究一般Kaehler流形上的扇形，我们考虑了扇形的弧长。在截面曲率从下而下的条件下，我们发现在复空间形式上弧的长度不长于相应扇形的弧的长度。我们还发现，当且仅当扇形是完全测地线和复杂嵌入扇形时，等式成立。(3) Kaehler磁流的力的变化和伪同余虽然我们有一个由Kaehler磁场的力的变化得到的真实的二维表面，但从映射的角度来看，它在一般Kaehler流形上并不那么自然。利用这一性质研究了复欧几里得因子Kaehler流形上Kaehler磁流的拟同余性。作为一种应用，我们用带环轨迹的Kaehler磁场的力来描述非紧化型厄米对称空间的秩。少

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