Editing Schur's theorem (section)

== Differential geometry ==

{{main|Bow lemma}}

In [[differential geometry]], '''Schur's theorem''' compares the distance between the endpoints of a space curve <math>C^*</math> to the distance between the endpoints of a corresponding plane curve <math>C</math> of less curvature.

Suppose <math>C(s)</math> is a plane curve with curvature <math>\kappa(s)</math> which makes a convex curve when closed by the chord connecting its endpoints, and <math>C^*(s)</math> is a curve of the same length with curvature <math>\kappa^*(s)</math>. Let <math>d</math> denote the distance between the endpoints of <math>C</math> and <math>d^*</math> denote the distance between the endpoints of <math>C^*</math>. If <math>\kappa^*(s) \leq \kappa(s)</math> then <math>d^* \geq d</math>.

'''Schur's theorem''' is usually stated for <math>C^2</math> curves, but [[John M. Sullivan (mathematician)|John M. Sullivan]] has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).