Unique Structure on G-equivariant Manifolds

TL;DR

This paper investigates the structure of G-equivariant maps from a Lie group to a manifold under group action, focusing on their computational and observability aspects, and explores conditions for a unique smooth structure.

Contribution

It introduces the concept of a unique smooth structure on the set of G-equivariant maps, enhancing understanding of their properties under group actions.

Findings

Identification of conditions for the existence of a unique smooth structure

Analysis of computational aspects of G-equivariant maps

Insights into observability related to equivariant map structures

Abstract

We will have a deep look at the set of all $G$-equivariant maps from the factor Lie group $G$ to the under the action manifold $M$, both from "computational" and "observability" viewpoint. We will also be looking for the existence of "unique" structure on this set, in a way that the induced-action of Lie group $G$ be smooth on the mentioned set.