Generalized pseudo-Riemannian geometry

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.