TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 11, Pages 4505–4523 S 0002-9947(02)03060-X Article electronically published on July 2, 2002 A BERGER-GREEN TYPE INEQUALITY FOR COMPACT LORENTZIAN MANIFOLDS MANUEL GUTI ´ ERREZ, FRANCISCO J. PALOMO, AND ALFONSO ROMERO 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. 1. Introduction In [13], L. W. Green solved the famous Blaschke conjecture in dimension two. A key tool to prove it was the Berger inequality, which asserts area(M,g) ≥ 2a 2 π χ(M ), χ(M ) being the Euler-Poincar´ e characteristic of M , for a 2-dimensional compact Riemannian manifold (M,g) without conjugate points before a fixed distance a in the parameter of any (unit) geodesic, and the equality holds only in case (M,g) has constant sectional curvature π 2 a 2 . This inequality was later generalized to higher dimensions by Berger and independently by Green [6, Proposition 5.64] as follows: Vol(M,g) ≥ a 2 π 2 n(n − 1)  M S dµ g , (1.1) where (M,g) is a compact Riemannian manifold of dimension n, scalar curvature S, and without conjugate points before a fixed distance a in the parameter of any geodesic. Moreover, equality holds if and only if (M,g) has constant sectional curvature π 2 a 2 . Note that the Berger inequality is a direct consequence from (1.1), via the Gauss-Bonnet theorem. The Berger-Green inequality can be equivalently written as follows: Vol(UM, ˆ g) ≥ a 2 π 2 (n − 1)  UM  Ric dµ ˆ g , (1.2) Received by the editors April 6, 2001 and, in revised form, April 11, 2002. 2000 Mathematics Subject Classification. Primary 53C50, 53C22; Secondary 53C20. Key words and phrases. Lorentzian manifolds, timelike conformal vector fields, null geodesics, conjugate points, Lorentzian odd-dimensional spheres. 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. c 2002 American Mathematical Society 4505 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use