This article is about mathematical objects that are uniquely determined by small amounts of information. For the mathematical study of rigid bodies, see
structural rigidity.
In
mathematics, a rigid collection C of
mathematical objects (for instance sets or functions) is one in which every c∈C is uniquely determined by less information about c than one would expect.
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.
Examples
Some examples include:
Harmonic functions 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,
polynomials 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. 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 ∈ L(X, Y) is completely determined by its values on any set of
basis vectors of X.
Mostow's rigidity theorem, which states that the geometric structure of negatively curved manifolds is determined by their topological structure.
Cauchy's theorem on geometry of
convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules.
In
combinatorics, the term rigid is also used to define the notion of a rigid surjection, which is a
surjection for which the following equivalent conditions hold:[1]
For every , ;
Considering as an -
tuple, the first occurrences of the elements in are in increasing order;
This relates to the above definition of rigid, in that each rigid surjection uniquely defines, and is uniquely defined by, a
partition of into pieces. Given a rigid surjection , the partition is defined by . Conversely, given a partition of , order the by letting . If is now the -ordered partition, the function defined by is a rigid surjection.
Structural rigidity, a mathematical theory describing the
degrees of freedom of ensembles of rigid physical objects connected together by flexible hinges.