Definition 4

If ψ is a φ-morphism of vector bundles of class C^r, E is called a vector G-bundle of class C^r.

In other words, E is a vector G-bundle of class C^r if for all (g_0, x_0) ∈ G × X the following condition holds: there exists an open neighbourhood U of (g_0, x_0) in G × X such that, if E' | U (resp. E' | φ(U)) is identified with a trivial vector bundle of fibre M (resp. N) by means of a vector chart, the mapping (g, x) ↦ ψ_{g, x} of U into L(M, N) is of class C^r.

The mapping ψ is obviously bijective and it follows from the above local criterion that ψ^{-1} is a φ^{-1}-morphism of vector bundles so that ψ is a φ-isomorphism of vector bundles.

A trivial vector G-bundle of base X is a vector bundle X × F (where F is a complete normable space) with the law of operation (g, (x, f)) ↦ (gx, f) of G on X × F.

We again assume the hypotheses and notation preceding Definition 4 and further take τ to be a vector functor of class C^r for isomorphisms (Differentiable and Analytic Manifolds, R, 7.6.6). Then τE is a vector bundle with base space X. For all x ∈ X, its fibre (τE)_x is equal to τ(E_x). For all normable spaces N_1, N_2, let Isom(N_1, N_2) denote the set of isomorphisms of N_1 onto N_2. If g ∈ G, then

The τ(ψ_{g, x}) define a law of left operation (g, u) ↦ gu of G on τE and the canonical projection of τE onto X is compatible with the operations of G on X and τE.