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.
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This is an example of what is known as a rigidity theorem : that in an extremal case of some invariant, we are able to conclude everything about the object.
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This fact is based on the Mostow rigidity theorem, that two isomorphic groups generated by reflections in H n for n > = 3, define congruent fundamental domains ( Vinberg polytopes ).
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Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their rigidity theorems of George Mostow and Grigory Margulis, and with Margulis arithmeticity.
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Liu's general rigidity theorem associated to loop groups generalizes Witten rigidity conjectures proved by Hirzebruch, Krichever, Landweber-Stong and Ochanine, which is the first time that Kac-Weyl character formula entered the fields of geometry and topology.
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The relevance of the hyperbolic geometry of a 3 & ndash; manifold to its topology also comes from the Mostow rigidity theorem, which states that the hyperbolic structure of a hyperbolic " & ndash; manifold of finite volume is uniquely determined by its homotopy type.
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He proved in that "'R "'4 has an uncountable number of smooth structures ( see also exotic "'R "'4 ), and ( with Raoul Bott in ) proved Witten's rigidity theorem on the elliptic genus.
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The Mostow rigidity theorem states that for lattices in simple Lie groups not locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ) ( the group of 2 by 2 matrices with determinant 1 ) any isomorphism of lattices is essentially induced by an isomorphism between the groups themselves.
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Throughout its history, the group has consisted of a diverse and dignified range of members, from poet Robert Frost to actor George Takei ( Sulu of Star Trek fame ) to G . D . Mostow the mathematician of Mostow rigidity theorem fame and Calvin Hill, NFL Rookie of the Year and multiple Pro Bowl selectee.
