Study On The Ideal Boundary Of Orbits And Hadamard K(?)hler Manifold

One of the most typical methods for studying manifolds is to characterize the shape and geometric structure of manifolds by examining the properties of curves.Since the curve on a manifold is a geodesic line when it is not affected by magnetic force,trajectory can be considered as a generalization of geodesic lines.In this paper,by investigating the properties of trajectory and using magnetic Jacobi fields,we obtain a comparison theorem for trajectory-spheres,and study the asymptotic behavior of ideal boundaries of Hadamard K(?)hler manifolds on a simply connected K(?)hler manifold with nonpositive sectional curvature.This thesis mainly does two works:The first work is to estimate the volume of the trajectory sphere in the K(?)hler magnetic field,according to the relation between the volume shape of the trajectory sphere and the magnetic Jacobi field,the upper bound estimate of the trajectory sphere is given by using the comparison theorem of the magnetic Jacobi field and the Rauch’s comparison theorem.Another work is to extend the existence and uniqueness of trajectory on Hadamard K(?)hler manifolds.Using orbitsHarps asymptotic behavior on Hadamard K(?)hler ideal boundary and cartan-Hadamard theorem,And proved that |k|≤(?),for any point and Hadamard K ddotahler manifolds ideal from any point on the border,there is only a track to connect these two points.