Decompositions of Analytic 1-Manifolds

TL;DR

This paper classifies analytic 1-manifolds based on their symmetry properties under Lie group actions, showing that free submanifolds decompose into countably many segments related by the symmetry group.

Contribution

It demonstrates that free analytic 1-manifolds are discretely generated by their symmetry group, extending previous classifications to a broader class of group actions.

Findings

Free analytic 1-manifolds decompose into countably many symmetry free segments.

Each segment is uniquely related by the Lie group action.

The decomposition holds under non-contractive group actions.

Abstract

In a previous article, 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 proven under the assumption that the action is non-contractive (which is less restrictive than regular and separately analytic).