Imersão estacionária de variedades Riemannianas no espaço-tempo FLRW esférico

Abstract

In this work, we study an extension of hu classical Takahashi’s theorem, thus giving a necessary and sufficient condition for a space-like immersion of a Riemannian manifold M in Ln+2 to be stationary (zero mean curvature vector) in I ×f Sn. For this we approach the main concepts of semi-Riemannian geometry which will be useful to us. We explore the isometric embeddeding of I ×f Sn into Ln+2, which allows us to contemplate I ×f Sn as a rotation hypersurface in Ln+2. After a detailed study of this hypersurface we can see any spacelike submanifold of I×f Sn as a spacelike submanifold of Ln+2. Finally, we study Takahashi’s theorem and Markvorsen’s theorem thus giving a historical context to arrive at the main theorem.