The Structure of the Internal Tangent Space to a Point of the Orbit Space of a Manifold under a Proper Lie Group Action
Isaac Cinzori
TL;DR
The thesis addresses the problem of understanding the internal tangent space $T_{[x]}(M/G)$ for a manifold $M$ under a proper Lie group action, when $M/G$ is viewed as a diffeological space. It builds a framework from diffeology and internal tangent spaces to Lie theory (including slice and tube theorems) and shows that $T_{[x]}(M/G)$ is isomorphic to the stratified tangent space via the orbit-type stratification, reducing computations to a compact linear action on a slice. The main result (ITS isomorphic to STS) provides a concrete bridge between diffeological and stratified viewpoints, enabling practical calculation through fixed-point subspaces $(T_xS)^{G_x}$. The work includes explicit examples with $\mathbb{R}/\text{O}(1)$ and $\mathbb{R}^3/\text{SO}(3)$, illustrating the reduction to a linear-algebraic problem and highlighting the potential for broader applications to diffeological analogues of manifold notions and orbifolds.
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.