Topics in Lorentzian and Finsler Geometry: geodesic flow and isometry group

• Grant number: 11/12565-7
• Support Opportunities: Scholarships in Brazil - Doctorate
• Effective date (Start): September 01, 2011
• Effective date (End): June 30, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Paolo Piccione
• Grantee: Leandro Augusto Lichtenfelz
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 07/03192-7 - Submanifold geometry and Morse theory in finite and infinite dimensions, AP.TEM

Abstract

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.