Recurrent Anholonomy in Curved Space Navigation Solved by the Riemann Zeta Function

Abstract

Navigation on curved surfaces has to handle additional nontrivial nonlinear geometric signals from (an)holonomy. Things may be even worse since propulsion recurrently couples back to the sensor signal. This effect can be experienced with orbiting gyroscopic sensors or spinning particles subject to spin-orbit coupling, which are just Gsensors of rotational acceleration. A solution to the rather simple orbital situation on spherically curved surfaces can be obtained by applying a Gudermannian mapping from precession patterns on the sphere onto holographic geometric phase patterns on the Poincaré disc by interference. This kind of stereographic mapping between surfaces of different constant curvatures induces the geometric extra rotations recurrently coupling back to the precession signal, where the fixed point solution to the emerging strange attractor is given by the functional equation of the Riemann zeta function.