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.
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.
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;
maps initial segments of
to initial segments of
.
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.
This article incorporates material from rigid on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.