Rigid

From Wikipedia, the free encyclopedia

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 C is uniquely determined by less information about c than is necessary a priori.

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:

1. Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
2. 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.
3. By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any countably 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.
4. Linear maps L(X,Y) between vector spaces X, Y are rigid in the sense that any L L(X,Y) is completely determined by its values on any set of basis vectors of X.
5. Mostow's rigidity theorem, which states that negatively curved manifolds are isomorphic if some rather weak conditions on them hold.
6. 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.
7. A 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.

This article incorporates material from rigid on PlanetMath, which is licensed under the GFDL.