Diffeomorphism invariant Colombeau algebras. Part III: Global theory

We present the construction of an associative, commutative algebra Ĝ(X) of generalized functions on a manifold X satisfying the following optimal set of permanence properties: (i) D(X) is linearly embedded into Ĝ(X), f(p) ≡ 1 is the unity in Ĝ(X). (ii) For every smooth vector field ξ on X there exists a derivation operator L̂ξ : Ĝ(X) → Ĝ(X) which is linear and satisfies the Leibniz rule. (iii) Lξ|D′(X) is the usual Lie derivative. (iv) ◦|C∞(X)×C∞(X) is the pointwise product of functions. Moreover, the basic building blocks of Ĝ(X) are defined in purely intrinsic terms of the manifold X.