论文标题
在爱因斯坦的条件下
On the Einstein condition for Lorentzian 3-manifolds
论文作者
论文摘要
众所周知，在洛伦兹几何形状中，没有紧凑的球形空间形式。在维度3中，这意味着没有封闭的爱因斯坦3型脉络膜具有阳性爱因斯坦常数。我们在这里概括了这一事实，证明也没有封闭的Lorentzian 3-manifolds $（m，g）$，其ricci张力满足$$ \ text {ric} = fg+（f-λ）t^{\ flat} \ flat} \ otimes t^{那永远不会占用$ 0，λ$。 （观察到，当$ f =λ$时，这将减少到阳性爱因斯坦案例。）我们表明，如果$λ$为负，则没有这种阻塞。最后，还检查了“边界”情况$λ= 0 $：我们表明，如果$λ= 0 $和$ f> 0 $，则$（m，g）$必须等于$（\ m athbb {s}^1 \！\！\！\！\！\！\！
It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds $(M,g)$ whose Ricci tensor satisfies $$ \text{Ric} = fg+(f-λ)T^{\flat}\otimes T^{\flat}, $$ for any unit timelike vector field $T$, any positive constant $λ$, and any smooth function $f$ that never takes the values $0,λ$. (Observe that this reduces to the positive Einstein case when $f = λ$.) We show that there is no such obstruction if $λ$ is negative. Finally, the "borderline" case $λ= 0$ is also examined: we show that if $λ= 0$ 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.
