Métricas críticas para funcionais cuadráticos da curvatura

Abstract

A central problem in pseudo-Riemannian geometry is the search for optimal me- trics with respect to a certain geometric property, which is often formalized in de- tecting critical metrics for a given functional. In this Ph. D. thesis we focus on di- mensions three and four to classify homogeneous Riemannian manifolds which are critical for some quadratic curvature functional. In addition, we construct metrics with non-constant scalar curvature which are critical for all these functionals simul- taneously. In the Lorentzian setting, critical metrics of dimension three are classified both in the homogeneous context and in the more general situation given by the condition of the curvature being modeled on a symmetric space. The analysis of critical metrics on Brinkmann waves allowed the construction of new solutions in several massive gravity theories.