论文标题
一般宇宙学模型的拓扑
The Topology of General Cosmological Models
论文作者
论文摘要
宇宙是有限的还是无限的，它具有什么形状？通常在宇宙学的标准模型的背景下研究了这些基本问题，其中相对较少，其中宇宙被认为是同质和各向同性的。在这里，我们在高度一般的宇宙学模型中解决了上述问题，唯一的假设是物质的平均流量是无关的。使用来自差异几何形状的技术，特别是帽子通用定理的扩展，我们得出了一种暗示有限宇宙并产生其直径的条件。此外，在涉及曲率和直径之间相互作用的较弱的条件下，假设宇宙是有限的（即具有封闭的空间切片），我们提供了可能的拓扑清单。即，空间部分将是环形拓扑$ s^1 \ times s^2 $，$ s^1 \ tilde {\ times} s^2 $，$ s^1 \ times \ times \ times \ mathbb {rp {rp}^2 $，$ \ mathb {rp}^3 \ $ s $ s $ s $ s $ s $或torus $ t^3 $。特别是，在这种情况下，将排除相关总和的基本结构（除一个），以及与负曲率相关的大量拓扑结构。这些结果是通过将几乎分裂定理的概括与EHLERS和ELLIS的曲率公式一起应用的概括，从3- manifolds的几何化的后果中获得。
Is the universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the universe is assumed to be homogeneous and isotropic. Here we address the above questions in highly general cosmological models, with the only assumption being that the average flow of matter is irrotational. Using techniques from differential geometry, specifically extensions of the Bonnet-Myers theorem, we derive a condition which implies a finite universe and yields a bound for its diameter. Furthermore, under a weaker condition involving the interplay between curvature and diameter, together with the assumption that the universe is finite (i.e., has closed spatial slices), we provide a concise list of possible topologies. Namely, the spatial sections then would be either the ring topologies $S^1 \times S^2$, $S^1\tilde{\times}S^2$, $S^1\times\mathbb{RP}^2$, $\mathbb{RP}^3 \# \mathbb{RP}^3$, or covered by the sphere $S^3$ or torus $T^3$. In particular, under this condition the basic construction of connected sums would be ruled out (save for one), along with the plethora of topologies associated with negative curvature. These results are obtained from consequences of the geometrization of 3-manifolds, by applying a generalization of the almost splitting theorem together with a curvature formula of Ehlers and Ellis.
