The geometry of weakly-Einstein manifolds

The geometry of Einstein manifolds is a well studied topic in differential geometry. Einstein metrics appear in the analysis of critical metrics for some Riemannian functionals. They add strong conditions on the manifold, so it is a natural question trying to weaken this condition. This thesis is devoted to the study of the so called weakly-Einstein manifolds. This conditions are analyzed under some special assumptions in different frames such as locally conformally flat manifolds, hypersurfaces or homogeneous manifolds. Moreover, other conditions related with the weakly-Einstein ones are also studied which are the generalized Einstein one and the two-loop renormalization flow.