Lorentzian manifolds with no null conjugate points
An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian $n(\geq3)$-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian $2-$torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.(Received November 27 2002)
(Revised April 2 2003)
p1 Current address: Departamento de Matemáticas Aplicada, Universidad de Málaga, 29071-Málaga, Spain.
1 Partially supported by MCYT-FEDER Grant BFM2001-1825.
2 Partially supported by MCYT-FEDER Grant BFM2001-2871-C04-01.