Rigid - Definition and Overview

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 one would expect.

It should emphasized that the above definition does not define a mathematical object. 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. 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.
3. 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.
4. Mostow's rigidity theorem

This article incorporates material from rigid (http://planetmath.org/?op=getobj&from=objects&id=621) on PlanetMath, which is licensed under the GFDL.