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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
Posted:
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:
http://dx.doi.org/10.1090/S0002-9947-02-03058-1
PII:
S 0002-9947(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
Posted:
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
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