# 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. 1.

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

2. 2.

By the fundamental theorem of algebra, polynomials in $\mathbb{C}$ 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. 3.

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

4. 4.

Mostow’s rigidity theorem

Title rigid Rigid 2013-03-22 14:38:10 2013-03-22 14:38:10 matte (1858) matte (1858) 11 matte (1858) Definition msc 00-01 rigidity result rigidity theorem rigidity