Rigidez e topologia do bordo para variedades (λ, n+ m)- Einstein generalizadas

Abstract

This dissertation presents a study on a class of compact manifolds with boundary that generalize Einstein manifolds. The main objective is to classify generalized (λ, n + m)- Einstein manifolds under certain hypotheses, which encompass well-known manifolds such as static manifolds and the static perfect fluid space-time. The results presented here are mainly based on one of the works by Allan Freitas and Márcio Santos, where the authors establish, from specific geometric conditions and with the help of classical theorems, such as the Bishop-Gromov Volume Comparison Theorem, a topological classification for the boundary of the generalized (λ, n + m)-Einstein manifolds and demonstrate rigidity properties for such manifolds.