Abstract.Let G be a connected Lie subgroup of the real orthogonal group O(ti). For the action of G on R", we construct linear subspaces a that intersect all orbits. We determine for which G there exists such an 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.Introduction. Let G be a compact Lie group acting on a real vectorspace V, and let ( • , -) be a G-invariant inner product on V. Having a linear cross-section a c V of minimal possible dimension (dim a = minxeKcodim{C7 • x}) can often be used to an advantage: In studying G-invariant differential equations it can be used for "reduction of variables" (see e.g. [4]); another obvious use is in analyzing the G-orbit structure of V. We show (Lemma 1) that such cross-sections always exist and how to construct them. In fact, this is extremely simple, the idea is to exploit the critical points of the function g-» (g-v,w), as was done by Hunt [6] to prove the conjugacy of Cartan subalgebras.The nicest situation arises when the G-orbits are orthogonal to the cross-section a. It is then natural to think of a and the G-orbits as giving polar coordinates on V, in analogy with the standard action of SO(n) on R". Therefore we call real representations of G, whose orbits admit orthogonal linear cross-sections, polar.If H/G is a symmetric space, then the action (here called symmetric space action) of G on the tangent space TeC to H/G at eG is polar [3]. It turns out that if w: G, -* O(V) is any polar representation, then there is a symmetric space H/G and a

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