Polar coordinates induced by actions of compact Lie groups

Abstract

Let G G be a connected Lie subgroup of the real orthogonal group O ( n ) O(n) . For the action of G G on R n {{\mathbf {R}}^n} , we construct linear subspaces a \mathfrak {a} that intersect all orbits. We determine for which G G there exists such an a \mathfrak {a} meeting all the G G -orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible G G arise as the isotropy subgroups of Riemannian symmetric spaces.