Symmetries of Analytic Curves

TL;DR

This paper classifies analytic curves and 1-submanifolds based on their symmetries under Lie group actions, revealing they are either exponential, decomposable into symmetry-free parts, or diffeomorphic to simple geometric objects.

Contribution

It provides a comprehensive classification of analytic curves and submanifolds by their symmetry properties, extending previous results to include 1-submanifolds and their decompositions.

Findings

Analytic curves are either exponential or split into countably many symmetry-generated segments.

Analytic 1-submanifolds are either free, exponential, or diffeomorphic to a circle or interval.

Decomposition results for free cases are outlined but proven separately.

Abstract

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic 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 discretely generated by the symmetry group (i.e., each such curve naturally decomposes into countably many symmetry free subcurves that are mutually and uniquely related by the Lie group action). We additionally extend the classification result to the analytic 1-submanifold case. Specifically, we show that an analytic 1-submanifold is either free or (exponential, i.e.) analytically diffeomorphic (via the exponential map) to the unit circle or an interval. The corresponding decomposition results in the free case are outlined in this paper, but proven in a separate one.