Diffeomorphism invariant Colombeau algebras. Part III: Global theory

Abstract

We present the construction of an associative, commutative algebra $\hat {\mathcal G}$ of generalized functions on a manifold $X$ satisfying the following optimal set of permanence properties: (i)The space of distributions on $X$ is linearly embedded into $\hat {\mathcal G}$, $f(p)\equiv 1$ is the unity in the algebra. (ii) For every smooth vector field $\xi$ on $X$ there exists a derivation operator $\hat L_\xi: \hat {\mathcal G} \to \hat {\mathcal G}$ which is linear and satisfies the Leibniz rule. (iii) $L_\xi$ restricted to the space of distributions on $X$ is the usual Lie derivative. (iv) Multiplication in the algebra restricted to the space of smooth functions is the usual (pointwise) product of functions. Moreover, the basic building blocks of $\hat {\mathcal G}$ are defined in purely intrinsic terms of the manifold $X$.