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:
 * 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 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.
 * 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 the geometric structure of negatively curved manifolds is determined by their topological structure.
 * 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) 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.
 * 8) Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface.
 * 9) Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups.
 * 10) 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:


 * 1) For every $$i, j \in m$$, $$i < j \implies \min f^{-1}(i) < \min f^{-1}(j)$$;
 * 2) Considering $$f$$ as an $$n$$-tuple $$\big( f(0), f(1), \ldots, f(n-1) \big)$$, the first occurrences of the elements in $$m$$ are in increasing order;
 * 3) $$f$$ 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.