Comparison on bow-shapes for Kaehler magnetic fields

研究人员的负责人主要研究由Kaehler磁场的轨迹自然获得的真实的二维表面。他比较了复空间形式上的这种曲面和一般Kaehler流形上的这种曲面。(1)弓形和新月形的比较给定一个Kaehler磁场的轨迹，我们在这个轨迹上的每个点上附加一条测地线，它的初始向量和轨迹的切线向量跨越切线空间中的一条复线。这样的测地线形成了一个真正的二维曲面。在复杂的空间形式上，这个表面是一条复杂的空间线。为了研究一般Kaehler流形上这个曲面的形状，对于这个轨迹的一部分，我们取这个曲面两端相连的一条测地线段。在截面曲率的条件下，我们发现测地线段的长度不小于复杂空间形式上的弓形长度。我们还发现，当且仅当新月…更多的是完全的测地线浸没的弓形。(2)扇形的比较我们考虑通过磁指数映射的复杂直线的图像。在复空间形式上，切复直线上的r-球的像是测地r-球与复空间直线的交集。为了在一般的Kaehler流形上研究这个扇形，我们考虑了这个扇形的弧长。在截面曲率自下而上的条件下，我们发现在复空间形式上，弧长不大于相应扇区的弧长。我们还发现，当且仅当扇区是完全测地线且复杂的嵌入扇区时，等式成立。(3)Kaehler磁流的力的变化和伪一致性虽然我们有一个由不同的Kaehler磁场力获得的真实的二维表面，但从映射的角度来看，它在一般的Kaehler流形上并不是那么自然。利用这一性质，我们研究了具有复欧氏因子的Kaehler流形上的Kaehler磁流的伪同余性质。作为应用，我们用Kaehler磁场的作用力刻画了非紧致型Hermitian对称空间的阶。较少

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