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Local rigidity of 3-dimensional cone-manifolds. (English) Zbl 1098.53038
Cone-manifolds can be viewed as generalizations of geometric orbifolds, where the cone-angles are no longer restricted to the set of orbifold-angles, which are rational multiples of $$\pi$$. In particular, it is known after W. P. Thurston that $$3$$-dimensional cone-manifolds arise naturally in the geometrization of $$3$$-dimensional orbifolds. In this paper, the author studies the local deformation space of $$3$$-dimensional cone-manifold structures of constant curvature $$\kappa\in\{-1,0,1\},$$ and cone-angles less than or equal to $$\pi.$$ In the hyperbolic and in the spherical cases his main result is a vanishing theorem for $$H^1_{L^2}(M;{\mathcal E})$$, the first $$L^2$$-cohomology group of the smooth part $$M=C\setminus\Sigma$$ of the cone-manifold $$C$$ ($$\Sigma$$ being the singular part of $$C$$), with coefficients in the flat bundle of infinitesimal isometries.
From this result, he can conclude local rigidity (in the spherical case it is assumed that $$C$$ is not Seifert fibered): the set of cone-angles $$\{\alpha_1,\ldots,\alpha_N\}$$, $$N$$ being the number of edges contained in $$\Sigma$$ (under the hypotheses $$\Sigma$$ turns out to be a trivalent graph), provides a local parametrization of the space of hyperbolic, resp. spherical, cone-structures near the given structure on $$M$$, and, in particular, there are no deformations that leave the cone-angles fixed. In the Euclidean case of the main theorem, the author proves that $H_{L^2}^1(M;{\mathcal E}_0)\simeq\{\omega\in\Omega(M,{\mathcal E}_0);\,\, \nabla\omega=0\},$ where $${\mathcal E}_0\subset{\mathcal E}$$ is the parallel sub-bundle of infinitesimal translations.

##### MSC:
 53C24 Rigidity results 57N65 Algebraic topology of manifolds 58D10 Spaces of embeddings and immersions
##### Keywords:
cone-manifolds; infinitesimal isometries; local rigidity
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