THE COVARIANT APPROACH TO SPACE TIMES ANDSOME EXTENDED THEORIES OF GRAVITY

The thesis deals with the covariant description of the geometry of some fundamental spacetimes and their realizations in two recent extensions of GR: Cotton and conformal Killing gravity. In Generalized Robertson - Walker spacetimes the Ricci tensor is the sum of a perfect fluid term and an anisotropic term given by the electric component of the Weyl tensor. In standard cosmology the anisotropy vanishes: this allows to specify the geometric correction due to the extended theory as a perfect fluid. The analysis is then extended to the broader class of doubly-warped spacetimes, employed in investigations of radiating stars and cosmological models. Here the Ricci tensor also contains a heat flow and anisotropic terms originating from the presence of a spatial acceleration vector. The geometric corrections in extended theories are investigated. In spherically symmetric static spacetimes the Faraday tensor is explained. General Relativity and extended theories coupled to linear or non-linear electrodynamics are explored. Cotton gravity was proposed to explain the shape of galaxy rotation curves without imposing dark matter. From the geometric point of view it is shown that the field equations can be recast into Einstein's ones augmented by an arbitrary Codazzi tensor. This simplifies the search of solutions. A new cosmological theory theory was proposed by J. Harada in 2023 to describe the effects of the presence of dark energy without assuming it. Remarkably, the field equations may be recast into Einstein equations augmented by an arbitrary divergence-free conformal Killing tensor, whereby the name conformal Killing gravity. With the new Friedmann equations the growth of the density contrast in a matter-dominated Universe is found to be very close to the \Lambda CDM behavior. The fit of the theoretical Hubble parameter H(z) with only Cosmic Chronometers or combined with Barion Acoustic Oscillations datasets, indicates the occurrence of a Big Rip or an exotic transition.