Polar coordinates induced by actions of compact Lie groups

Dadok, Jiri

doi:10.1090/S0002-9947-1985-0773051-1

Polar coordinates induced by actions of compact Lie groups
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Trans. Amer. Math. Soc. 288 (1985), 125-137 Request permission

Abstract:

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

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