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Generalized pseudo-Riemannian geometry
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In this paper, generalized tensor analysis is employed to introduce a nonlinear distributional pseudo-Riemannian geometry, and the notion of geodesics of a generalized metric is defined.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.read more
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References
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Book
Semi-Riemannian Geometry With Applications to Relativity
TL;DR: In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.
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Singular hypersurfaces and thin shells in general relativity
TL;DR: In this article, an approach to study the dynamics of thin shells of dust in general relativity is presented. But no mention of admissible or even any space-time co-ordinates is needed.
Book
Spinors and space-time
Roger Penrose,Wolfgang Rindler +1 more
TL;DR: The calculus of 2-spinors was introduced and systematically developed in this article, which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations.
Book
The Convenient Setting of Global Analysis
Andreas Kriegl,Peter W. Michor +1 more
TL;DR: Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly real compact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifold, infinite dimensional differential geometry Manifolds of Mappings Further applications References as mentioned in this paper.