A non-smooth theory for sub-Lorentzian geometry

Objective

"The project SUBLOR aims to establish the foundations of sub-Lorentzian geometry, the non-holonomic analogue of Lorentzian geometry, within the new synthetic theory of Lorentzian length spaces and metric spacetimes. While recent developments have extended Lorentzian geometry to the non-smooth setting, spacetimes with non-holonomic constraints have been overlooked, despite their central role in modern geometry. Sub-Riemannian geometry has flourished into a thriving field of geometric analysis, whereas its Lorentzian counterpart has seen little development. SUBLOR will address this gap by: (1) Integrating sub-Lorentzian structures into the framework of Lorentzian length spaces and metric spacetimes. Sub-Lorentzian geometry will be defined in the general control-theoretic setting, its causal structure and topologies analysed, and Pontryagin's maximum principle applied to the study of geodesics, showing that sub-Lorentzian manifolds are Lorentzian length spaces. (2) Characterising the infinitesimal geometry of sub-Lorentzian manifolds. The timelike metric tangents will be shown to be quotients of sub-Lorentzian Carnot groups, and a Lorentzian version of the Ball-Box theorem (a ""Diamond-Box"" theorem) will be established. (3) Developing a theory of curvature for sub-Lorentzian geometry. The optimal-transport-based timelike curvature-dimension conditions and their variants will be investigated, and the more intrinsic Hamiltonian curvature will be introduced through Jacobian fields and expansions of the time-separation function. By building these three foundational layers, which also underpin the success of sub-Riemannian geometry, SUBLOR will systematise sub-Lorentzian geometry, highlight a novel class of spacetimes, and open the way for future applications in higher-dimensional models, quantum gravity, and analogue gravity."