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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
Email:
mgl@agt.cie.uma.es
Francisco J. Palomo
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos, 29071 Málaga, Spain
Email:
fpalo1@clientes.unicaja.es
Alfonso Romero
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain.
Email:
aromero@ugr.es
DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03060-X
PII:
S 0002-9947(02)03060-X
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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