Calculus and Ricci curvature in non-smooth semi-Riemannian geometry

Thesis

This thesis presents several results that connect the synthetic approach—via optimal transport on metric measure spaces—with the analytic approach—via distributional methods on low-regularity manifolds—in the study of nonsmooth semi-Riemannian geometry, encompassing both Riemannian and Lorentzian metric tensors of low regularity.

In a preliminary Chapter 2, we collect relevant definitions and background that the results in this thesis require.

In Chapter 3, we present the results from a collaboration with Andrea Mondino, where we established the equivalence of distributional and synthetic lower Ricci curvature bounds for weighted Riemannian manifolds with continuous metric tensor that admits L 2 loc-Christoffel symbols and a weight in C 0 ∩ W 1,2 loc .

In Chapter 4, we study infinitesimal Hilbertianity (namely, the property that the synthetic Sobolev space W1,2 w is Hilbert) on manifolds endowed with a discontinuous Riemannian metric g. In dimensions d ≥ 3, we provide examples where g, g−1 ∈ L ∞ loc ∩ W 1,p loc for p ∈ [1, d − 1] which fail to be infinitesimally Hilbertian.

In Chapter 5, we prove that strongly causal and causally simple spacetimes arising from manifolds with continuous Lorentzian metric tensors are infinitesimally Minkowskian, the Lorentzian analogue of infinitesimal Hilbertianity.

Authors: Ryborz, VE

• DOI: 10.5287/ora-ozrqadnrx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: differential geometry; geometric analysis
• Subjects: Mathematics