The global geometry of manifolds endowed with a semi-Riemannian (i.e.,m non positive definite) metric tensor is quite different from the Riemannian case. For instance, compact Lorentz manifolds may fail to be geodesically complete or geodesically connected. Moreover, their isometry group may fail to be compact.On the other hand, Lorentz manifolds that admitr a somewhere timelike Killing vector field show some properties similar to the Riemannian case, for instance, they are geodesically complete, and they always admit non trivial closed geodesics. In this research project we will try to establish some Riemannian-like properties of compact stationary Lorentz manifolds, like for instance:Conjecture 1.: the isometry group of a simply connected compact stationary Lorentz manifold is compact;Conjecture 2: Compact stationary Lorentz manifolds are geodesically connected.Conjecture 3.: Compact stationary Lorentz manifolds have discrete spectrum.
News published in Agência FAPESP Newsletter about the scholarship: