On the Geometry of the Set of Orbits of Killing Vector Fields on Euclidean Space

Abstract

Smooth orbifolds can be considered as a natural generalization of smooth manifolds, when model space is not ${\mathbb R}^n,$ but factor space $ {\mathbb R}^n/G$ by a finite group $G$ of diffeomorphisms. In modern theoretical physics, orbifolds are used as string propagation spaces and are motivated by the preference for models on orbifolds over models on ordinary manifolds. Orbifolds arise in the theory of foliations as ``good'' spaces of leaves. It is known that the existence of a proper leaf with a finite holonomy group for a transversally complete Riemannian foliation is a necessary and sufficient condition for the space of leaves to be an orbifold. The purpose of our paper is study the structure of the space of leaves of singular foliation which generated by orbits of Killing vector fields. Geometry of Killing vector fields is a object of many investigations due to their importance in geometry and physics. We will prove that the space of orbits of family of Killing vector fields on Euclidean space is a smooth orbifold.