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The first result of this paper is that the total Gaussian curvature of a compact Lorentz manifold vanishes (Theorem I, sec. 8). The argument in the proof is different from those given in [1] and [2]. In fact, we make extensive use of the existence of a global line field on a Lorentz manifold, rather than reducing the problem to the classical case of a Riemannian manifold. In §4 the index of...