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Generalized pseudo-Riemannian geometry. (English) Zbl 1012.46049

Authors’ 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.

MSC:

46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
46T30 Distributions and generalized functions on nonlinear spaces
46F10 Operations with distributions and generalized functions
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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