2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68457We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. We prove this by finding the exact solutions of the corresponding system of partial differential equations.10 pagesDifferential GeometryMetric Geometry53A04text