Concerning this question, approached from the analytical side, several fundamental contributions have appeared recently. More precisely, we shall be interested in the problem of finding low-regularity extensions of spacetimes. In this paper, we will concentrate on a generalisation of Lorentzian geometry suitable for the low-regularity setting. In the synthetic direction, the theory of Alexandrov spaces with curvature bounded above and/or below is well-developed as an appropriate generalisation of Riemannian geometry with sectional curvature bounds (see, for instance, ), and the work of Lott–Villani–Sturm gives a generalisation of the notion of a Riemannian metric with lower bound on the Ricci curvature to metric measure spaces . Examples in this direction in the Lorentzian setting are the positive mass theorem for distributional curvature, work on cone structures and the recent work of extending the classical singularity theorems to C 1, 1-regularity, which in turn builds on previous results in low-regularity Lorentzian geometry and causality. In the context of low-regularity Riemannian geometry, examples of a result of an analytical nature would be DeTurck and Kazdan’s study concerning harmonic coordinates , Taylor’s results on regularity of isometries and Lytchak and Yaman’s result that minimising curves for C 0, α Riemannian manifolds are C 1, β curves, where β = α 2 - α. Here, curvature bounds for Alexandrov spaces and CAT( k) spaces are defined in terms of comparison properties of geodesic triangles. The other approach to studying low-regularity geometries is by “synthetic” or metric space methods. For example, one can study geometrical properties of (pseudo-)Riemannian metrics that have regularity C 0, C 0, α or C 1, 1, etc., or so-called Geroch–Traschen metrics, for which the Christoffel symbols are L loc 2, and the curvature is well-defined as a distribution . One approach is analytical, where one lowers the differentiability assumptions on, for example, (pseudo-)Riemannian metrics below the level where curvature can be classically defined. One can distinguish between two main lines of research in low-regularity geometry.
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