2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217058A short proof of the Caratheodory conjecture about index of an isolated umbilic on the convex 2-dimensional sphere is suggested. The argument is based on the study of geodesic lines near cone-type singularity of a metric induced by holomorphic quadratic differentials.Differential Geometry30F30; 53A07; 58F10text