Kaeler magnetic fields and Carnot spaces

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The head investigator studied trajectories for Kaehler magnetic fields on symmetric spaces. His works, a part of which is a joint work with some coinvestigators, can be classified into the following four directions.(1) Mean operators associated with magnetic fieldsIn order to study the relationship between trajectories for Kaehler magnetic fields and Schroedinger operators, the head investigator studied magnetic random walks. On complex space forms the mean operator generated by this magnetic random walk has the same properties as that generated by the geodesic random walk. On the other hand, if we studied a magnetic spherical mean which is derived from potential on unit tangent bundle we found the principal term of the formal expansion was the Schroedinger operator.(2) Circles on complex space formsExtending the notion of trajectories for Kaehler magnetic fields the head investigator studied length spectrum for circles on complex space forms. The moduli space of circles on these space … More s are open rectangles in a Euclidean plane parametrized by geodesic curvature and complex torsion. Concerning the continuity of length spectrum for circles we found it had a natural foliation structure. By use of this structure we clearfied set theoretic properties of length spectrum, the asaymptotic behavior of the number of congruency classes of circles with respect to their length, and the properties of the k-th length spectrum function with respect to the geodesic curvature.(3) Geodesics on geodesic spheres in a rank one symmetric spacesHaving been inspired with the idea in our study on circles we studied lengths of geodesics on a geodesic sphere in a rank one symmetic space, which is famous as an example of Berger sphere. We considered geodesics on a geodesic sphere as curves on a complex space form, and studied their horizontal lifts with respect to the Hopf fibration. We could then treat them as curves in a Euclidean space. We showed the relationship between the radius of a geodesic sphere and length-simplicity of geodesics and clearfied the asymptotic behavior of the number of closed geodesics on a geodesic sphere with respect to their length.(4) Characterizations of submanifolds in complex space formsBy use of properties of circles and helices on complex space forms the head investigator and S. Maeda characterized submanifolds in complex space forms. Their idea stands on the technique of treating geodesic and circles on a submanifold as curves on a complex space form. They characterized homogeneous submanifold, Veronese embeddings and some other important submanifolds. Less