Um estudo sobre a curvatura escalar e curvatura média de ordem superior de hipersuperfícies tipo-espaço em espaços-tempos GRW

Abstract

In this work, we obtain new results for a space-like hypersurface immersed in a generalized Robertson-Walker spacetime (GRW) to be a slice, under the assumption of higher-order average curvature. For this, we calculate the operator Lk acting on the height function h and on the function ⟨N,K⟩. Using the formula found for the operator Lk acting on the height function, we were able to develop successive Minkowski-type integral formulas for compact space-like hypersurfaces immersed in a GRW spacetime, which are essential tools to obtain and extend results from uniqueness to such hypersurfaces in the GRW. In a second moment, we study the scalar curvature of space-like hypersurfaces immersed in an arbitrary GRW space-time and then apply the results obtained to understand the behavior of the scalar curvature of space-like hypersurfaces immersed in space-time. More specific GRWs, like the Einsteins and Parabolics.