Ruled real surfaces formed by Kaehler magnetic fields

在研究黎曼流形时，测地线无疑是一个重要的研究对象。但是如果我们考虑所有光滑曲线的族，测地线族是一个小族。因此，首席研究员通过研究卡勒磁场的轨迹来研究卡勒流形，卡勒磁场是卡勒形式的常数倍。比较定理为了研究可变全纯截面曲率的Kaehler流形，我们考虑由轨迹和轨迹扇区构成的直纹实曲面上的新月。在对Kaehler流形截面曲率的假设下，我们可以用复空间形式2上对应对象的长度来估计这些对象的回路长度。复杂空间形式下测地线球上佐佐木磁场的轨迹我们考虑奇维流形上的佐佐木磁场。这对应于实际四维流形上的凯勒磁场。更复杂的空间形式是模型空间，但Sasaki磁场的轨迹性质与Kaehler磁场在复杂空间形式上的轨迹性质有很大不同。有些轨迹长度相同，但彼此不相等。一些Kaehler流形的等长浸没表征我们通过等长浸没到真实空间形式来研究Kaehler流形。由于等距浸入在流形上具有一定的结构刚性，我们考虑具有2阶点的曲线族，其中包括Kaehler磁场的轨迹族。我们可以将完全脐带浸没或第一标准浸没的复杂空间形式描述为其诱导映射保持2阶性质和曲率对数导数的空间形式。如果我们将2阶性质的条件弱化为其外在形状为2阶的条件，那么我们就可以刻画秩小于3的厄米对称空间

When we study Riemannian manifolds, it is needless to say that geodesics play quite important object. But if we consider the family of all smooth curves, the family of geodesics is a small family. In this reason the head investigator studied Kaehler manifolds by investigating trajectories for Kaehler magnetic fields, which are constant multiples of the Kaehler form1.Comparison theoremsIn order to study Kaehler manifolds of variable holomorphic sectional curvatures, we consider crescents on a ruled real surface formed by trajectory and trajectory-sectors. Under an assumption on sectional curvatures of a Kaehler manifold, we can estimate lengths of circuits of these objects by length of corresponding objects on a complex space form.2.Trajectories for Sasaki magnetic fields on geodesic spheres in a complex space formWe consider Sasaki magnetic fields on odd dimensional manifolds. This corresponds to Kaehler magnetic fields on real eavn dimensional manifolds. Though geodesic spheres in com … More plex space forms are model spaces, properties of trajectories for Sasaki magnetic fields are quite different from properties of trajectories for Kaehler magnetic fields on complex space forms. There are trajectories which have the same length but are not congruent to each other.3.Characterizations of some Kaehler manifolds through isometric immersionsWe study Kaehler manifolds through isometric immersions into real space forms. Since isometric immersions give some structural rigidity on manifolds, we consider the family of curves having points of order 2, which includes the family of trajectories for Kaehler magnetic fields. We can characterize complex space forms immersed by totally umbilic immersions or 1st standard embeddings as those whose induced maps preserve order 2 property and curvature logarithmic derivative. If we weaken the condition on order 2 property to the condition that their extrinsic shapes are of order 2, then we can characterize Hermitian symmetric spaces of rank less than 3 Less