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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


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.

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