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.
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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 $i, j \in m$, $i < j \implies \min f^(i) < \min f^(j)$;
# 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;
# $f$ 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 $n = f^(0) \sqcup \cdots \sqcup f^(m-1)$. Conversely, given a partition of $n = A_0 \sqcup \cdots \sqcup A_$, order the $A_i$ by letting $A_i \prec A_j \iff \min A_i < \min A_j$. If $n = B_0 \sqcup \cdots \sqcup B_$ 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.

See also

*Uniqueness theorem
*Structural rigidity, a mathematical theory describing the degrees of freedom of ensembles of rigid physical objects connected together by flexible hinges.
*Level structure (algebraic geometry)

References

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Mathematical terminology