Extension of actions on Stiefel manifolds
Isabel Dotti Miatello · Pacific Journal of Mathematics · 1979
It is natural to ask for examples of π-biaxial actions in the unitary and simplectic case which do not come from the orthogonal case.Here we provide examples of such actions.Introduction* Let us consider the left translation action ofThe main result to be proved here is that the above action can not be extended to a biaxial O(2n)(U(2n)) action.The proof uses strongly the correspondence between U(n)(Sp(n)) π-biaxial manifolds with orbit space diffeomorphic to a disk and framed submanifolds of the sphere.The main references for this article are the book Introduction to Compact Transformation Groups (Bredon [3]) for the general theory of groups actions and the mimeographed notes (Bredon [4]) Biaxial Actions of the Classical Groups for the classification of such actions and characterization of restrictions of ττ-biaxial manifolds.1* Preparatory material* Let G be a compact Lie group and σ: G -> Gl{ V) be a representation of G on the real vector space V.By a G-manifold "modeled on σ" we mean a smooth G manifold such that each orbit in M has an open invariant neighborhood which is equivariantly diffeomorphic to an open invariant set in the representation space V oί σ.Let d = 1, 2, 4. In these three cases we let Gί stand for O(n), U(n) or Sp(n).The standard representation of Gί on R nd will be denoted by σ n and the trivial real fc-dimensional representation byDEFINITION 1.1.A Gί manifold M, n^2, will be called biaxial if it is modeled on the representation 2σ n + θ k .It is not hard to see that a Gί manifold is biaxial iff the following four conditions hold:1.The principal orbit type is Gί\Gί^.The other orbit types (if any) are Gί\Gί_ x and fixed points (if any).2. The representation of Gί about a fixed point is 2σ n + θ k .3. The slice representation of G Li on the normal space to an orbit at a point with isotropy group Gί_± is σ n _ x + θ k +d+i-