Draft:Bow lemma

Bow lemma, or Schur's theorem — a theorem in differential geometry of curves. It compares the distance between the endpoints of a space curve $C^{*}$ to the distance between the endpoints of a corresponding plane curve $C$ of less curvature.

Formulation

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

Schur's theorem is usually stated for $C^{2}$ curves, but there are versions for piecewise smooth curves and curves of finite total curvature.

See also

• Schur's lemma (Riemannian geometry)

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Categories:

• Theorems in differential geometry