Mostow rigidity theorem

The Mostow rigidity theorem states that finite-volume hyperbolic manifolds of dimension at least three with isomorphic fundamental groups are isometric. Their hyperbolic geometry is therefore fixed by their topology.

Hyperbolic manifold: A manifold equipped with a Riemannian metric of constant negative sectional curvature. The theorem concerns finite-volume examples of these spaces in dimensions at least three.
Riemannian manifold: A smooth manifold equipped with a smoothly varying inner product on its tangent spaces. Hyperbolic manifolds are Riemannian manifolds with constant negative curvature.
G. D. Mostow: An American mathematician known for foundational work on rigidity of lattices and hyperbolic manifolds. His 1968 work established the rigidity result now bearing his name.
Mostow–Prasad rigidity for lattices: A rigidity result stating that isomorphisms between suitable lattices in rank-one Lie groups arise from ambient isometries. The manifold theorem reflects a broader principle about discrete groups acting on hyperbolic space.
Teichmüller space: A parameter space of marked conformal or hyperbolic structures on a surface, modulo equivalence. Unlike higher-dimensional hyperbolic manifolds, surfaces admit continuous families of hyperbolic metrics.
Fundamental group: A group encoding the homotopy classes of loops based at a point in a topological space. An isomorphism between these groups is the theorem’s topological input.
Sectional curvature: The Gaussian curvature of a two-dimensional tangent plane in a Riemannian manifold. Hyperbolic geometry is characterized by constant negative sectional curvature.
Mostow–Prasad rigidity: The rigidity theorem for finite-volume hyperbolic manifolds, including noncompact cases, associated with Mostow and Prasad. Prasad extended the theorem to include finite-volume noncompact manifolds.
Hyperbolic volume: The volume measured by a hyperbolic metric on a hyperbolic manifold. For finite-volume manifolds in the theorem, the hyperbolic metric and its volume are topologically determined.
Uniformization theorem: The theorem that every simply connected Riemann surface is conformally equivalent to the sphere, plane, or disk. Uniformization classifies conformal structures on surfaces rather than making each surface’s hyperbolic metric unique.

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Mikhail GromovRelated: Gromov’s geometric viewpoint helped shape the broader study of rigidity phenomena.