We prove that a non-totally vicious n-dimensional compact spacetime $(M,g)$ admitting a parallel lightlike vector field is foliated by compact totally geodesic null hypersurfaces. As a consequence, assuming non-negative Ricci curvature on the leaves then the first Betti number of M is bounded above by n with equality if and only if M is diffeomorphic to the torus.

中文翻译：

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具有平行光似矢量场的时空的第一个Betti数

我们证明了一个非完全恶性的n维紧时空$（\mathrm{中号}，G）$准平行光样矢量场被紧凑的全测地零超曲面所掩盖。结果，假设叶子上的Ricci曲率为非负值，则当且仅当M对圆环是变态时，M的第一个Betti数由n限定在n之上。