Geometry Research Group at the University of São Paulo

The Geometry Research Group at IME USP

Events

Mini-curso: Pós Graduação do Programa do Mat-IME-USP.

Course:	An introduction to causality theory of space-times, focusing on classification results of globally hyperbolic constant curvature space-times and conformally flat ones.
Professor:	Prof:Clara Rossi Salvemini ( Université de Paris-Sud ,Paris 11)
Organized by:	Prof. Paolo Piccione
Date and place:	23 a 27 de fevereiro, das 10hs às 12hs

Description:Causal theory of space-times have been mainly developped during 70's by mathematical physicists to produce models of the space-time in accord with the equations of general relativity. The new conception brought by general relativity, where the space and time are not absolut and indipendent but are depending by the observator, made unuseful the mathematical models of univers based on the Euclidean space, where the time is just a direction like the others. On a Lorentzian manifold there is a natural way to differenciate courbes, by the signe of the Lorentian ''norme'' of their tangent vector. In this way it's possible to look at the space-time as a single whole in which it is already encoded which are the admissible trajectories for particules, which are not, and, above all, which are the trajectories of the light. In others words Lorentzian manifolds are naturally endowed with some kind rigid geometric structure, in good agreement with the needs of the physics. From a mathematical point of vue, this sort of natural structure of Lorentzian manifolds, called the causal structure, is something rich and fascinating to study in itself. In these lectures we will focus on globally hyperbolic space-times. This is the stronger hypothesis that one can put over the causal structure of a space-time, usually in order to solve the evolution problem associated to the Einstein equation. It tourns out that constant curvature globally hiperbolic space-times are particular solutions for this problem (into the void). These space-time have been classified (starting from '90), from a geometric point of view, when the spacelike-slices are compact or, in some cases, even just complete (seen as riemannian manifolds). We will give an idea of the proof of some of these results which involves tools of (G,X)-structures theory, dynamic of group action on manifolds, hyperbolic geometry, and conformally flat geometry, and causality.