On the geometry of the orbits of Killing vector fields

TL;DR

This paper investigates the geometric structure of foliations generated by orbits of Killing vector fields on manifolds, providing a complete classification in the specific case of three-dimensional Euclidean space.

Contribution

It offers a full geometric classification of foliations formed by Killing vector field orbits on R^3, advancing understanding of their structure.

Findings

Complete geometric classification of Killing orbit foliations in R^3

Analysis of the structure of singular foliations generated by Killing fields

Insights into the geometry of orbits of symmetries on manifolds

Abstract

Let $D$ be a set of smooth vector fields on the smooth manifold $M$.It is known that orbits of $D$ are submanifolds of M. Partition $F$ of M into orbits of $D$ is a singular foliation. In this paper we are studying geometry of foliation which is generated by orbits of a family of Killing vector fields.In the case $M=R^3$ it is obtained full geometrical classification of $F$. Throughout this paper the word "smooth" refers to a class $C^\infty$.