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Deformation spaces on geometric structures. (English) Zbl 0798.53030
Matsumoto, Y. (ed.) et al., Aspects of low dimensional manifolds. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 20, 263-299 (1992).
This paper is devoted to the theory of deformation spaces on geometric structures. A basic theorem of Thurston asserts that there is a canonical decomoposition by convex hulls on a hyperbolic surface \(S\) which admits a (one dimensional complex) projective structure. This result is generalized to conformally flat structures on manifolds in arbitrary dimensions. A \((G,X)\)-structure (geometric structure) on a smooth \(n\)- manifold is a maximal collection of charts modelled on a simply connected \(n\)-dimensional homogeneous space \(X\) of a Lie group \(G\) whose coordinate changes are restrictions of transformations from \(G\). The authors examine the structure of the deformation space of \((G,X)\)-structures invariant under Lie groups. Finally, they give a description of the deformation space of \(S^ 1\) invariant spherical CR structures and \(S^ 1\) invariant conformally flat structures.
For the entire collection see [Zbl 0771.00015].

53C10 \(G\)-structures
57M50 General geometric structures on low-dimensional manifolds
53C20 Global Riemannian geometry, including pinching