Decompositions of analytic 1-manifolds

Abstract

In an author’s previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author’s previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action.