On Thurston's Geometrical Space Form Problem: On Quasi Space Forms

Year 2024, , 232 - 244, 23.04.2024

Stefan Haesen Miroslava Petrović-torgašev , Leopold Verstraelen

https://doi.org/10.36890/iejg.1466330

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 inWeyl’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.

Keywords

Quasi space forms, Riemannian geometry, Deszcz symmetric spaces

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