Symmetries of Analytic Paths

TL;DR

This paper classifies the symmetries of analytic paths in manifolds under Lie group actions, distinguishing between continuous and discrete symmetries, and characterizes these symmetries using reparametrization sets related to Lie subgroups of O(2).

Contribution

It provides a complete classification of path symmetries in analytic manifolds, linking them to Lie subgroups of O(2) through reparametrization sets.

Findings

Paths are either integral curves of fundamental vector fields or decomposable into translates of free segments.

Reparametrization sets are characterized by Lie subgroups of O(2).

Continuous symmetries correspond to infinite subgroups, discrete symmetries to finite subgroups.

Abstract

The symmetries of paths in a manifold $M$ are classified with respect to a given pointwise proper action of a Lie group $G$ on $M$. Here, paths are embeddings of a compact interval into $M$. There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $M$ (continuous symmetry). Secondly, paths that can be decomposed into finitely many pieces, each of which is the translate of some free segment, where possibly the translate is cut at the two ends of the paths (discrete symmetry). Here, a free segment is a path $e$ whose $G$-translates either equal $e$ or intersect it in at most finitely many points. Note that all the statements above are understood up to the parametrization of the paths. We will show, for the category of analytic manifolds, that each path is of exactly one of either types. For the proof, we use that the overlap of a path $\gamma$ with one of its translates is encoded uniquely in a mapping between subsets of $\dom\gamma$. Running over all translates, these mappings form the so-called reparametrization set to $\gamma$. It will turn out that, up to conjugation with a diffeomorphism, any such set is given by the action of a Lie subgroup of $O(2)$ on $S^1$, restricted in domain and range to some compact interval on $S^1$. Now, the infinite subgroups correspond to the continuous symmetry above, finite ones to the discrete symmetry.