New Geometry in Classical and Quantum Gravity

In this thesis, we predominantly study variants on the theory of general relativity. In order to do this in a mathematically rigorous way, we introduce the theories of pseudo-Riemannian geometry, Cartan geometry and quantum Riemmannian geometry. Next, we prove that the same concepts from differential geometry in those different theories, such as a metric, connection, torsion, curvature an a action, are equivalent.

After that, we use Cartan geometry, Spencer cohomology in particular, to describe the cokernel of the Spencer differential for Galilean and Carrollian p-branes and to give geometric criteria for the different cases for which the intrinsic torsion of a given spacetime reside in one to the subrepresentations. We subsequently repeat those results in the language of pseudo-Riemannian geometry (with indices) and give a number of examples of Galilean limits of general relativity, for which we see that the intrinsic torsion can reside in those subrepresentations. We also do this for Newton-Cartan geometry and string Newton-Cartan geometry, obtaining comparable results.

Furthermore, we introduce quantum Riemannian geometry to study Euclidean discrete models. We give the equations which a *-compatible quantum Levi-Civita connection of such a model must satisfy, and we give a solution for metrics that are invariant in transversal directions. At last, we classify 2-term L∞-algebras in terms of a Lie algebra, a representation of that Lie algebra, a cohomology class of the corresponding Lie algebra cohomology and a vector space.