rigid

Suppose $C$ is a collection of mathematical objects (for instance, sets, or functions). Then we say that $C$ is rigid if every $c\in C$ is uniquely determined by less information about $c$ than one would expect.

It should be emphasized that the above “definition” does not define a mathematical object. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians.

Let us illustrate this by some examples:

1. Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.

2. 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 countably infinite set, say $\mathbb{N}$, or the unit disk.

3. Linear maps $ℒ(X,Y)$ between vector spaces $X,Y$ are rigid in the sense that any $L\in ℒ(X,Y)$ is completely determined by its values on any set of basis vectors of $X$.

4. Mostow’s rigidity theorem

Title	rigid
Canonical name	Rigid
Date of creation	2013-03-22 14:38:10
Last modified on	2013-03-22 14:38:10
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	11
Author	matte (1858)
Entry type	Definition
Classification	msc 00-01
Synonym	rigidity result
Synonym	rigidity theorem
Synonym	rigidity