Rigidity (mathematics)

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.