In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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:

1. Harmonic function
   Harmonic function
   In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
   
   s 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
   Schwarz lemma
   In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove...
   
    is an example of such a rigidity theorem.
3. By the fundamental theorem of algebra
   Fundamental theorem of algebra
   The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
   
   , polynomial
   Polynomial
   In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
   
   s 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. Note that 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 negatively curved manifolds are isomorphic if some rather weak conditions on them hold.
6. A well-ordered set is rigid in the sense that the only (order-preserving) automorphism
   Automorphism
   In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
   
    on it is the identity function. Consequently, an isomorphism
   Isomorphism
   In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...
   
    between two given well-ordered sets will be unique.
7. Cauchy's theorem
   Cauchy's theorem (geometry)
   Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...
   
    on geometry of convex polytope
   Convex polytope
   A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...
   
   s states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules.

See also

• Structural rigidity
  Structural rigidity
  In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.-Definitions:...
  
  , a mathematical theory describing the degrees of freedom
  Degrees of freedom (statistics)
  In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
  
  of ensembles of rigid physical objects connected together by flexible hinges.