It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds $(M,g)$ whose Ricci tensor satisfies $$ \text{Ric} = fg+(f-\lambda)T^{\flat}\otimes T^{\flat}, $$ for any unit timelike vector field $T$, any positive constant $\lambda$, and any smooth function $f$ that never takes the values $0,\lambda$. (Observe that this reduces to the positive Einstein case when $f = \lambda$.) We show that there is no such obstruction if $\lambda$ is negative. Finally, the "borderline" case $\lambda = 0$ is also examined: we show that if $\lambda = 0$ and $f > 0$, then $(M,g)$ must be isometric to $(\mathbb{S}^1\!\times \!N,-dt^2\oplus h)$ with $(N,h)$ a Riemannian manifold.

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关于洛伦兹三流形的爱因斯坦条件

众所周知，在洛伦兹几何中，没有紧凑的球面空间形式；在维度 3 中，这意味着不存在具有正爱因斯坦常数的封闭爱因斯坦 3-流形。我们在这里概括这个事实，通过证明也没有闭洛伦兹三流形 $ (M, g) $ 的 Ricci 张量满足 $$ \ text {Ric} = fg + (f- \ lambda) T ^ {\ flat } \ otimes T ^ {\ flat}，$$ 用于任何单位类时间向量场$ T $，任何正常数$ \ lambda $，以及任何从不取值0, \ lambda $的平滑函数$ f $ . （请注意，当 $ f = \ lambda $ 时，这将简化为正爱因斯坦情况。）我们表明，如果 $ \ lambda $ 为负，则没有这种障碍。最后，还检查了“边界”情况 $\lambda = 0 $：我们证明，如果 $\lambda = 0 $ 且 $f>0 $，则 $(M,g)$ 必须与 $(\mathbb {S} ^ 1 \! \ 次 \! N,