Symmetries of Analytic Curves

Abstract

Analytic curves are classified w.r.t. their symmetries under a regular Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves; each of them decomposing naturally into symmetry free subcurves mutually and uniquely related by the group action. We conclude that a connected analytic 1-dimensional submanifold is either analytically diffeomorphic to the unit circle or some interval, or that each point (except for at most countably many) admits a symmetry free chart.Comment: 49 page