Topological rigidity  In the mathematical field of topology, a manifold "M" is called topologically rigid if every manifold homotopically equivalent to "M" is also homeomorphic to "M". A central problem in topology is determining when two spaces are the same i.e. homeomorphic or diffeomorphic. Constructing a morphism explicitly is almost always impractical. If we put further condition on one or both spaces (manifolds) we can exploit this additional structure in order to show that the desired morphism must exist. Rigidity theorem is about when a fairly weak equivalence between two manifolds (usually a homotopy equivalence) implies the existence of stronger equivalence homeomorphism, diffeomorphism or isometry. A closed topological manifold "M" is called topological rigid if any homotopy equivalence "f" : "N" → "M" with some manifold N as source and M as target is homotopic to a homeomorphism. Example 1. If closed 2-manifolds "M" and "N" are homotopically equivalent then they are homeomorphic. Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism. Example 2. If a closed manifold "M""n" ("n" ≠ 3) is homotopy-equivalent to "S""n" then Mn is homeomorphic to "S""n". A diffeomorphism of flat-Riemannian manifolds is said to be affine iff it carries geodesics to geodesic. If "f" : "M" → "N" is a homotopy equivalence between flat closed connected Riemannian manifolds then "f" is homotopic to an affine homeomorphism. Theorem: Let "M" and "N" be compact, locally symmetric Riemannian manifolds with everywhere non-positive curvature having no closed one or two dimensional geodesic subspace which are direct factor locally. If "f" : "M" → "N" is a homotopy equivalence then "f" is homotopic to an isometry. Theorem (Mostow's theorem for hyperbolic "n"-manifolds, "n" ≥ 3): If "M" and "N" are complete hyperbolic "n"-manifolds, "n" ≥ 3 with finite volume and "f" : "M" → "N" is a homotopy equivalence then "f" is homotopic to an isometry.