# Rigidity (mathematics)

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.
* 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.
* 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.
* Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface.
* Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups.
* Rigid groups in the inverse Galois problem.

## Combinatorial use

In combinatorics, the term rigid is also used to define the notion of a rigid surjection, which is a surjection $f:n\to m$ for which the following equivalent conditions hold:
* For every $i,j\in m$, $i<j\implies \min f^{-1}(i)<\min f^{-1}(j)$;
* Considering $f$ as an $n$-tuple ${\bigf(0),f(1),\ldots ,f(n-1){\big )}$, the first occurrences of the elements in $m$ are in increasing order;
* $f$ maps initial segments of $n$ to initial segments of $m$.
This relates to the above definition of rigid, in that each rigid surjection $f$ uniquely defines, and is uniquely defined by, a partition of $n$ into $m$ pieces. Given a rigid surjection $f$, the partition is defined by $n=f^{-1}(0)\sqcup \cdots \sqcup f^{-1}(m-1)$. Conversely, given a partition of $n=A_{0}\sqcup \cdots \sqcup A_{m-1}$, order the $A_{i}$ by letting $A_{i}\prec A_{j}\iff \min A_{i}<\min A_{j}$. If $n=B_{0}\sqcup \cdots \sqcup B_{m-1}$ is now the $\prec$-ordered partition, the function $f:n\to m$ defined by $f(i)=j\iff i\in B_{j}$ is a rigid surjection.