Ruled real surfaces formed by Kaehler magnetic fields

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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