Line fields and Lorentz manifolds
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  and . 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 a line field at an isolated singularity is classical case of a Riemannian manifold. sum of a line field on a compact Riemannian 4-manifold in terms of the total Gaussian curvature. A consequence of this result is that every compact 4-manifold which admits a line field, also admits a vector field without zeros. (Theorem III, sec. 13). Finally, in sec. 15 it is shown that the notions of orientability and time-orientability on a Lorentz manifold are independent.
KeywordsVector Field Riemannian Manifold Vector Bundle Line Field Fibre Bundle
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