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A Berger-Green type inequality for compact Lorentzian manifolds

Authors: Manuel Gutiérrez, Francisco J. Palomo and Alfonso Romero
Journal: Trans. Amer. Math. Soc. 354 (2002), 4505-4523
MSC (2000): Primary 53C50, 53C22; Secondary 53C20
Published electronically: July 2, 2002
Erratum: Trans. Amer. Math. Soc. (recently posted)
MathSciNet review: 1926886
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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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Additional Information

Manuel Gutiérrez
Affiliation: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, 29071 Málaga, Spain

Francisco J. Palomo
Affiliation: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, 29071 Málaga, Spain

Alfonso Romero
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain.

Keywords: Lorentzian manifolds, timelike conformal vector fields, null geodesics, conjugate points, Lorentzian odd-dimensional spheres.
Received by editor(s): April 6, 2001
Received by editor(s) in revised form: April 11, 2002
Published electronically: July 2, 2002
Additional Notes: The first author was partially supported by MCYT-FEDER Grant BFM2001-1825, and the third author by MCYT-FEDER Grant BFM2001-2871-C04-01.
The second author would like to dedicate this paper to the memory of his grandmother Pepa.
Article copyright: © Copyright 2002 American Mathematical Society

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