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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $G = \\mathrm{SO}_0(n, 1)$ be the group of orientation-preserving isometries of hyperbolic space $\\mathbb{H}^n$.\nLet $\\Gamma \\subset G$ be a discrete torsion-free subgroup such that the hyperbolic manifold $M = \\Gamma \\backslash \\mathbb{H}^n$ is compact.\nThen the inclusion $\\Gamma \\to G$ is locally rigid.\nThat is, let $\\rho \\colon \\Gamma \\to G$ be a [[homomorphism|0JYTG5K9Y4]] close enough to the inclusion on a finite generating set of $\\Gamma$.\nThen $\\rho$ is conjugate to the inclusion by an element of $G$.\nEquivalently, the hyperbolic structure of $M$ admits no nontrivial small deformations.\n\n## Remarks\n\nCalabi proved this for compact hyperbolic manifolds, and Weil extended it to all uniform lattices in semisimple groups.\nIt follows from [[Weil's local rigidity theorem|V2HTMG4KT1]], since $\\Gamma$ is a uniform [[lattice|BTYF5Y5PFY]] in $G$, which for $n \\ge 3$ is simple, has trivial center and is not locally isomorphic to $\\mathrm{SL}_2(\\R)$.\nThe hypothesis $n \\ge 3$ is necessary: for $n = 2$ the hyperbolic structures on a closed surface of genus $g \\ge 2$ form a [[Teichmüller space|ZZGC6BZNJ5]] of real dimension $6g - 6$.\nFor $n = 3$ compactness is also needed: hyperbolic structures on cusped $3$-manifolds deform to incomplete structures, which give hyperbolic Dehn filling.\n\n## Explanations\n\n### L1 human\n\nA closed surface of genus two can carry many different hyperbolic shapes, which can be changed continuously.\nIn dimension three and higher this freedom disappears.\nA closed hyperbolic manifold cannot be bent slightly into a different hyperbolic manifold.\nIts shape is locally fixed by its topology.\n\n### L2 human\n\nThis theorem is the local version of [[Mostow rigidity|4BEWKR0EG7]], which states that the hyperbolic structure is determined by the fundamental group globally.\nIn the language of representations, the point of $\\Hom(\\Gamma, G)/G$ defined by $M$ is isolated.\nThe proof goes through the vanishing of the cohomology group $H^1(\\Gamma, \\mathfrak{g})$ with coefficients in the Lie algebra of $G$.\n",
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