An integral inequality on compact Lorentz manifolds, and its applications

Abstract

Bochner's technique is shown to be useful on compact Lorentz manifolds. It is proved that all the compact Einstein Lorentz manifolds admitting a timelike Killing vector field have non-positive scalar curvature, and the flat ones are (up to a covering) those isometric to a Lorentzian «-torus. Thus several results by Kamishima [4] are widely extended. Other classification results, including applications to the homogeneous case and non-existence consequences, are obtained. 1

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