Decompositions of Analytic 1-Manifolds

Abstract

In a previous paper, connected analytic 1-dimensional submanifolds with boundary have been classified w.r.t. their symmetry under a given regular Lie group action on an analytic manifold. It was shown that each such submanifold is either free or analytically diffeomorphic to the unit circle or some interval via the exponential map. In this paper, we show that each free connected analytic 1-submanifold naturally splits into symmetry free segments, mutually and uniquely related by the group action. This is proven under the assumption that the action is non-contractive, which is even less restrictive than regularity.Comment: 25 page