Trajectory-harps and Horns Applied to the Study of the Ideal Boundary of a Hadamard Kähler Manifold

Abstract

A trajectory-harps is made of a trajectory for a Kähler magnetic field $\mathbb{B}_{\kappa}$ and an associated variation of geodesics, and a trajectory-horn is made of a geodesic and an associated variation of trajectories. On a Hadamard Kähler manifold $M$ we study thickness and string-angles of trajectory-harps, and study tube-lengths and tube-angles of trajectory-horns. As an application of these we show that two distinct points on the compactification of $M$ with geometric ideal boundary can be joined by a trajectory for $\mathbb{B}_{\kappa}$ if the strength $|\kappa|$ is less than the upper bound of sectional curvatures of $M$.