Sobre alguns aspectos geométricos da gravitação: mergulhos do espaço- tempo em dimensões superiores, a conjectura de Wheeler e variedades de Weyl

Abstract

In this thesis we present several embedding problems related to modern theories about space-time. First we deal with the so called thin sandwich conjecture proposed by J. A. Wheeler. We show that the Bartnik-Fodor theorem extends naturally to higher dimensions and that, furthermore, the geometric hypotheses needed for the proofs can always be satisfied on compact manifolds. From this result, we conclude that on any compact n-dimensional manifold, n ≥ 3, there is an open set in the space of possible initial data where the thin sandwich problem is well-posed. Then, we apply this result to prove that any compact n-dimensional Lorentzian manifold, n ≥ 3, with metric in the Sobolev space Hs+3, s > n 2, has an embedding in a (2n+2)-dimensional Ricci-flat semi-Riemannian manifold. This result improves the codimension needed for the embedding quite drastically compared with previously known results. Finally, we study some embedding problems within the context of Weyl’s geometry. With respect to this problem, we show that the Campbell-Magaard theorem naturally extends to the Weyl integrable case, but, in the general case we find some no-go results and, as a very particular case, for the embed ding of 3-dimensional Weyl structures, there is as an analogue of the Campbell-Magaard theorem. We conclude by analysing the initial value formulation of a particular class of geometric scalar-tensor theories of gravity, where we will show that the vacuum case is well-posed.