Generalized pseudo-Riemannian geometry

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.

23 pages, Latex, final form

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Cited by in corpus (2)

• Nonlinear distributional geometry and general relativity
• Generalized functions valued in a smooth manifold