A global theory of algebras of generalized functions

Abstract

We present a geometric approach to defining an algebra $\hat{\mathcal{G}}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal{D}}^{\prime }(M)$ of distributions on $M$. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of $\hat{\mathcal{G}}(M)$. $\hat{\mathcal{G}}(M)$ is a{\em differential} algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ${\mathcal{D}}^{\prime }(M)$ into $\hat{\mathcal{G}}(M)$ that renders ${\mathcal{C}}^{\infty }(M)$ a faithful subalgebra of $\hat{\mathcal{G}}(M)$. Finally, it is shown that this embedding commutes with Lie derivatives. Thus $\hat{\mathcal{G}}(M)$ retains all the distinguishing properties of the local theory in a global context.

24 pages, LaTeX

References in corpus (2)

• Geodesics for impulsive gravitational waves and the multiplication of distributions
• On the foundations of nonlinear generalized functions I

Cited by in corpus (5)

• Generalised hyperbolicity in space-times with conical singularities
• Symmetry group analysis of weak solutions
• Nonlinear distributional geometry and general relativity
• Report on GRG18, Session A3, Mathematical Studies of the Field Equations
• Group analysis of differential equations and generalized functions