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

TL;DR

The paper introduces quasi space forms (QCC spaces) as Riemannian manifolds equipped with two orthogonal distributions whose tangent-plane curvatures depend only on their relative position to the distributions. It develops the framework of Deszcz symmetric spaces, Weyl conformal geometry, and quasi Einstein spaces to connect isotropy-breaking geometries with Thurston's non-isotropic model geometries. The main contributions show that conformally Euclidean quasi Einstein spaces are exactly quasi space forms, that in dimension three QE spaces coincide with QCC, and that conformally Euclidean Deszcz symmetric spaces are either real space forms or quasi space forms. These results unify several symmetric-space classes under the umbrella of quasi space forms and illuminate their role in Thurston-type geometries and conformal geometry.

Abstract

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.

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Theorem H
• Theorem I
• Theorem J