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Generalized pseudo-Riemannian geometry


Authors: Michael Kunzinger and Roland Steinbauer
Journal: Trans. Amer. Math. Soc. 354 (2002), 4179-4199
MSC (2000): Primary 46F30; Secondary 46T30, 46F10, 83C05
Published electronically: June 3, 2002
MathSciNet review: 1926870
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Abstract: Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-) Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.


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Additional Information

Michael Kunzinger
Affiliation: Department of Mathematics, University of Vienna, Strudlhofg. 4, A-1090 Wien, Austria
Email: Michael.Kunzinger@univie.ac.at

Roland Steinbauer
Affiliation: Department of Mathematics, University of Vienna, Strudlhofg. 4, A-1090 Wien, Austria
Email: roland.steinbauer@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9947-02-03058-1
Keywords: Algebras of generalized functions, Colombeau algebras, generalized tensor fields, generalized metric, (generalized) pseudo-Riemannian geometry, general relativity.
Received by editor(s): August 9, 2001
Received by editor(s) in revised form: January 31, 2002
Published electronically: June 3, 2002
Additional Notes: This work was in part supported by research grant P12023-MAT of the Austrian Science Fund
Article copyright: © Copyright 2002 American Mathematical Society