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Regenerating hyperbolic and spherical cone structures from Euclidean ones. (English) Zbl 0897.58042
From the author’s abstract: “We show that, in some cases, an Euclidean cone structure on a closed \(3\)-manifold can be deformed into hyperbolic or spherical cone structures by moving the singular angle. We describe other deformations by using generalized Dehn surgery parameters. To do that, we study the relationship between algebraic deformations of the holonomy representation and of the geometric structure.” In addition, a local formula for the volume of the deformed spaces near the Euclidean one’s is proven. Outside the singular set, the space in question has a nice, but incomplete, (hyperbolic/Euclidean,…) structure. The holonomy of such spaces is studied.

58H15 Deformations of general structures on manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57M50 General geometric structures on low-dimensional manifolds
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