The Structure of the Internal Tangent Space to a Point of the Orbit Space of a Manifold under a Proper Lie Group Action

TL;DR

This paper investigates the internal tangent space at points in the orbit space of a manifold under a proper Lie group action, revealing its isomorphism to the stratified tangent space in the diffeological setting.

Contribution

It provides a detailed description of the internal tangent space in orbit spaces and establishes its isomorphism with the stratified tangent space, bridging diffeological and stratified perspectives.

Findings

Internal tangent space description for orbit spaces

Isomorphism between internal and stratified tangent spaces

Extension of tangent space concepts to diffeological orbit spaces

Abstract

A diffeological space is a set equipped with a smooth structure, known as a diffeology, which allows us to extend certain notions from manifolds to these more general spaces. We study a generalized notion of tangent space to a point of a manifold, namely the internal tangent space to a point of a diffeological space. In particular, we study these internal tangent spaces when the diffeological space in question is the orbit space of a manifold acted upon by a proper Lie group action. We provide a useful description for an arbitrary internal tangent space to a point of such an orbit space and then, in the culmination of our work, show that the internal tangent space to a point of an orbit space, viewed as a diffeological space, is isomorphic to the stratified tangent space to the same point, when the orbit space is viewed as a stratified space with the well-known orbit type stratification.