A Unified Approach to Geometric Modifications of Gravity

This thesis studies modified theories of gravity from a geometric viewpoint. We review the motivations for considering alternatives to General Relativity and cover the mathematical foundations of gravitational theories in Riemannian and non-Riemannian geometries. Then, starting from the decomposition of the Einstein-Hilbert action into bulk and boundary terms, we construct new modifications of General Relativity. These modifications break diffeomorphism invariance or local Lorentz invariance, allowing one to bypass Lovelock's theorem while remaining second-order and without introducing additional fields. In the metric-affine framework, we introduce a new Einstein-Cartan-type theory with propagating torsion. Important comparisons are made with the modified teleparallel theories, and we construct a unified framework encompassing all these theories. The equivalence between theories that break fundamental symmetries in the Riemannian setting and non-Riemannian theories of gravity is explored in detail. This leads to a dual interpretation of teleparallel gravity, one in terms of geometric quantities and the other in terms of non-covariant objects. We then study the cosmological applications of these modified theories, making use of dynamical systems techniques. One key result is that the modified Einstein-Cartan theories can drive inflation in the early universe, replacing the initial cosmological singularity of General Relativity. To conclude, we discuss the viability of these modifications and possible future directions, examining their significance and relevance to the broader field of gravitational physics.