Generalized pseudo-Riemannian geometry

Kunzinger, Michael; Steinbauer, Roland

doi:10.1090/S0002-9947-02-03058-1

Generalized pseudo-Riemannian geometry
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by Michael Kunzinger and Roland Steinbauer PDF

Trans. Amer. Math. Soc. 354 (2002), 4179-4199 Request permission

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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