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

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.