论文标题
紧凑型排名折叠的拓扑
The topology of compact rank-one ECS manifolds
论文作者
论文摘要
具有平行的Weyl张量的伪 - 河 - 歧管歧管，它们不是形式平坦的或局部对称的，也称为EC歧管，具有自然的局部不变，等级等于1或2，并且是某个特定杰出的null null平行分布分布$ \，\ Mathcal {D} $的维度。所有已知的紧凑型EC歧管示例均为排名第一，并且具有大于4的尺寸。我们证明，紧凑的等级 - 一级ec歧管（即使不是本地同质）在必要时被两倍的等距覆盖物所取代，必须是$ \，\ scal {D}^perp $ severs os fibers os fibers os n fibers of $ \，\ \ pers of $。如果一个人假设$ \，\ Mathcal {d}^\ perp $至少有一个紧凑的叶子，则在本地均匀的情况下同样结论。我们还表明，在任何紧凑的等级的ECS歧管的伪里曼尼亚通用覆盖空间中，$ \，\ Mathcal {d}^\ perp $的叶子是全球产品分解的因子歧管。
Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as ECS manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the dimension of a certain distinguished null parallel distribution $\,\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a two-fold isometric covering, must be a bundle over the circle with leaves of $\,\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally-homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold the leaves of $\,\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.
