Lie Group Actions

Abstract

This chapter discusses the differentiable actions of Lie groups and their orbits. The model for orbits are quotient spaces G∕H. When H is closed, the quotient G∕H admits a structure of differentiable manifold, which was built in Chapter 6. One of the present objectives is then to verify that an orbit G ⋅ x is an immersed submanifold diffeomorphic to the quotient space G∕G _x, where G _x is the isotropy subgroup at x, which is a closed subgroup. In this direction, a convenient point of view is to look at orbits as maximal integral manifolds of a singular distribution (see Appendix B). This approach provides the additional information that they are quasi-regular immersed submanifolds.