Equivalence of Nearby Differentiable Actions of a Compact Group

In this note we will be concerned with the proof and consequences of the following fact : if o is a differentiate action of a compact Lie group on a compact differentiate manifold M, then any differentia t e action of G on M sufficiently close to 0o in the C^topology is equivalent to 0O. 1. Notation. In what follows differentiate means class C. If M and V are differentiate manifolds, 2HI(M, V) is the space of differe n t i a t e maps of M into V in the C^-topology where K is a positive integer or <*> fixed throughout. We denote by Diff (M) the group of automorphisms of M topologized as a subspace of 2iïl(Af, M). As such it is a topological group. £>(ilf) is the subgroup of Diff (M) consisting of diffeomorphisms which are the identity outside of some compact set and £>o(M) is the arc component of iM, the identity map of M, in SD(ikf). If M is compact £>(Af) is locally arcwise connected and 2Do(M) is open in £>(lf) and in fact in 2HX(Af, M). For a definition of the C^-topology and a proof of the statements made above, see [ô]. If G is a Lie group we denote by Œ(G, M) the space of differentiate actions of G on M, i.e. continuous homomorphisms of G into Diff (M), topologized with the compact-open topology. If ' g-^g* is an element of d(G, M) then by a theorem of D. Montgomery [2] 4>: (g, m)—^gm is an element of 2fTC(GXM, M). Given 0£Œ(G, M) and/GDiff (M) then composed with the inner automorphism of Diff (M) defined by ƒ is another element f(j> of a(G, M)(g*=fg*f-). Clearly (ƒ, <ƒ>)->ƒ<£ is jointly continuous and defines an action of Diff (M) on Œ(G, M). We henceforth consider a(G, M) as a Diff (M)-space and, a fortiori as a SD(M) and SD0(Af)space. Note that the orbit space Œ(G, M)/Diff (M) is just the set of equivalence classes of actions of G on M.