A Berger-Green type inequality for compact Lorentzian manifolds

Gutiérrez, Manuel; Palomo, Francisco; Romero, Alfonso

doi:10.1090/S0002-9947-02-03060-X

A Berger-Green type inequality for compact Lorentzian manifolds
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by Manuel Gutiérrez, Francisco J. Palomo and Alfonso Romero

Trans. Amer. Math. Soc. 354 (2002), 4505-4523

DOI: https://doi.org/10.1090/S0002-9947-02-03060-X

Published electronically: July 2, 2002

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Erratum: Trans. Amer. Math. Soc. 355 (2003), 5119-5120.

Abstract:

We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

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