A G-version of Smale's theorem

Abstract

We will prove the equivariant version of Smale&#39;s transversality theorem: suppose that the compact Lie-group G acts on the compact differentiable manifold M on which an invariant Morse-function f and an invariant vector field X are given so that X is gradient-like with respect to f (i.e. X(f)<0 away from critical orbits and X is the gradient of f (w.r.t. a fixed invariant Riemannian metric) on some invariant open subsets about critical orbits of f.) Given a bound $ε>0$ we will prove the existence of an invariant vector field Y of class C^1 for which vector field X+Y is also gradient-like such that: (a) |Y|_1<ε(here |.|_1 is the C^1 norm). (b)The intersection of the stable and unstable sets of vector field X+Y taken at a pair of critical orbits of f is transverse when restricted to an orbit type of the action.