Some results on differentiable actions

In this note, we shall announce some results on differentiable actions of SO(n), SU(w) and Sp(w) on manifolds. Since the detailed proofs are too long to be included here, we shall publish them elsewhere. THEOREM 1. Let <j> be a differentiable action of SO(n), (S\J(n), Sp(w)) on an m-dim manifold M where w e l l and m ^ ( « l ) 2 / 4 (n^8 and mS(n — l)/2, n^S and m^(n — l)). If the first rational Pontrjagin class of M, Pi(M), vanishes, then the identity component of any isotropy subgroup, (Gx)ofor xGM m is always conjugate to SO(fe), (SU(fe), Sp(&)) under the standard inclusion for some k^\n. THEOREM 2. For a given differentiable action <j> of SO(n), (SU(w), SpOO) on a homotopy sphere S {respectively Euclidean space R, respectively disc D) where n^ll and mS(n — l)/4: (n*£& and m^(n — l)/2, n^S and mS(n — l)) , we have that (i) all orbits are real (complex, quaternionic) Stiefel manifolds, (ii) if SO(n)/SO(k), (SU(n)/SXJ(k), Sp(#)/Sp(fe)) is the principal orbit and F is the fixed point set, then #*(F; A)~H*(Sy, A) (respectively H*(F] A) ~ H*(Rv; A) respectively H*(F\ A) ~ #*(£? ; A))