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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].

##### MSC:
 53C10 $$G$$-structures 57M50 General geometric structures on low-dimensional manifolds 53C20 Global Riemannian geometry, including pinching