On geodesic envelopes and caustics

Abstract

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along the tangent geodesics). Stable singularities of tangential caustics and geodesic envelopes are discussed. We also prove the (global) stability of tangential caustics of close curves in convex closed surfaces under small deformations of the initial curve and of the ambient metric.