标题： 关于汤姆森的几何空间形式问题：关于拟空间形式 显示英文标题

标题： On Thurston's geometrical space form problem: on quasi space forms

摘要： 提出了一种可能最为基本的黎曼空间的建议，这些空间是齐性但非各向同性的。 换句话说：提出了一种可能最为优美的对称空间的建议，超越了实数空间形式，即超越了齐性和各向同性的黎曼空间。 上述“最优美对称空间”的限定在某种程度上得到了验证，因为这些空间与实数空间形式一起，在人类视觉中我们能够轻易识别共形事物的重要性方面，以及在威尔的观点中，黎曼几何中的对称性方面，都是最自然的。 遵循他去除实数空间形式各向同性条件的建议，因此引入的准空间形式为 Thurston 在其 1979 年的普林斯顿讲义《三维流形的几何与拓扑》中提出的几何空间形式问题提供了一种度量的、局部的几何解法。 粗略地说，准空间形式是维数大于或等于 3 的黎曼流形，它们不是实数空间形式，但它们可以容纳两个正交互补的分布，使得在所有点上，所有在切空间中相对于这些分布处于相同位置的 2-平面都具有相同的截面曲率。 显示英文摘要

摘要： A proposal is made for what may well be the most elementary Riemannian spaces which are homogeneous but not isotropic. In other words: a proposal is made for what may well be the nicest symmetric spaces beyond the real space forms, that is, beyond the Riemannian spaces which are homogeneous and isotropic. The above qualification of `'nicest symmetric spaces'' finds a justification in that, together with the real space forms, these spaces are most natural with respect to the importance in human vision of our ability to readily recognise conformal things and in that these spaces are most natural with respect to what in Weyl's view is symmetry in Riemannian geometry. Following his suggestion to remove the real space forms' isotropy condition, the quasi space forms thus introduced do offer a metrical, local geometrical solution to the geometrical space form problem as posed by Thurston in his 1979 Princeton Lecture Notes on `'The Geometry and Topology of 3-manifolds''. Roughly speaking, quasi space forms are the Riemannian manifolds of dimension greater than or equal to 3, which are not real space forms but which admit two orthogonally complementary distributions such that at all points all the 2-planes that in the tangent spaces there are situated in a same position relative to these distributions do have the same sectional curvatures.