Mostow’s rigidity theorem and its generalization. An outline of the proof
Let V* and V be compact locally symmetric spaces of nonpositive curvature with isomorphic fundamental group. Hence V* = X*/Γ*, V = X/Γ, where X* and X are symmetric spaces and Γ* is isomorphic to Γ. Let us assume that in the de Rham decomposition of X* and X there are no Euclidean factors and no factors isometric to the hyperbolic plane, then by the famous rigidity theorem of [Mostow, 1973], V* and V are isometric up to normalizing constants. Thus, if the metric of X is multiplied on each de Rham factor by a suitable constant, then X*/Γ* and X/Γ are isometric.
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