Rigid 1458875 216070628 2008-05-30T23:10:59Z Algebraist 266905 remove superfluous, misleading word {{otheruses4|mathematics|the materials sense|Stiffness}} {{wiktionarypar|Rigid}} In [[mathematics]], suppose ''C'' is a collection of mathematical objects (for instance sets or functions). Then we say that ''C'' is '''rigid''' if every ''c'' <math>\in</math> ''C'' is uniquely determined by less information about ''c'' than one would expect. It should be emphasized that the above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians. Some examples include: #[[Harmonic function]]s on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. #[[Holomorphic functions]] are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The [[Schwarz lemma]] is an example of such a rigidity theorem. #By the [[fundamental theorem of algebra]], [[polynomial]]s in '''C''' are rigid in the sense that any polynomial is completely determined by its values on any [[infinite set]], say '''N''', or the [[unit disk]]. Note that by the previous example, a polynomial is also determined within the set of holomorphic functions by the finite set of its non-zero derivatives at any single point. #Linear maps '''L'''(''X'',''Y'') between vector spaces ''X'', ''Y'' are rigid in the sense that any '''L''' <math>\in</math> '''L'''(''X'',''Y'') is completely determined by its values on any set of [[basis vector]]s of ''X''. #[[Mostow's rigidity theorem]], which states that negatively curved manifolds are isomorphic if some rather weak conditions on them hold. #A [[well-ordered set]] is rigid in the sense that the only ([[order-preserving]]) [[automorphism]] on it is the identity function. Consequently, an [[isomorphism]] between two given well-ordered sets will be unique. #A [[Euclidean group|rigid motion]] of a subset of [[Euclidean space]] is not always defined the same: it may be any distance-preserving transformation of the collection of points (i.e. a composition of translations, rotations, and reflections), or only those preserving orientation (i.e. a composition of translations and rotations). In the latter case the concept of rigidity is analogous to that of a physically inflexible solid, which must be moved as a single entity so that its movement (up to atomic motions indiscernible to the naked eye) is completely determined by the displacement of a single "point" and the orientation of the solid body about that point. More generally, a rigid motion of a metric space is a (self)-[[isometry]]. {{planetmath|id=6219|title=rigid}} [[Category:Mathematical terminology]] [[nl:Rigiditeit]]