New York Journal of Mathematics
Volume 18 (2012) 139-199


Michael Grosser, Michael Kunzinger, Roland Steinbauer, and James A. Vickers

A global theory of algebras of generalized functions. II. Tensor distributions

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Published: March 19, 2012
Keywords: Tensor distributions, algebras of generalized functions, generalized tensor fields, Schwartz impossibility result, diffeomorphism invariant Colombeau algebras, calculus in convenient vector spaces
Subject: Primary 46F30; Secondary 46T30, 26E15, 58B10, 46A17

We extend the construction of the authors' paper of 2002 by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby obtain a universal algebra of generalized tensor fields canonically containing the space of distributional tensor fields. The canonical embedding of distributional tensor fields also commutes with the Lie derivative. This construction provides the basis for applications of algebras of generalized functions in nonlinear distributional geometry and, in particular, to the study of spacetimes of low differentiability in general relativity.


The first author was supported by FWF grants P23714, Y237 and P20525 of the Austrian Science Fund

Author information

Michael Grosser:
University of Vienna, Faculty of Mathematics

Michael Kunzinger:
University of Vienna, Faculty of Mathematics

Roland Steinbauer:
University of Vienna, Faculty of Mathematics

James A. Vickers:
School of Mathematics, University of Southampton