Английская Википедия:Fenchel's theorem
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In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least <math>2\pi</math>. Equivalently, the average curvature is at least <math> 2 \pi/L</math>, where <math>L</math> is the length of the curve. The only curves of this type whose total absolute curvature equals <math>2\pi</math> and whose average curvature equals <math> 2 \pi/L</math> are the plane convex curves. The theorem is named after Werner Fenchel, who published it in 1929.
The Fenchel theorem is enhanced by the Fáry–Milnor theorem, which says that if a closed smooth simple space curve is nontrivially knotted, then the total absolute curvature is greater than Шаблон:Math.
Proof
Given a closed smooth curve <math>\alpha:[0,L]\to\mathbb{R}^3</math> with unit speed, the velocity <math>\gamma=\dot\alpha:[0,L]\to\mathbb{S}^2</math> is also a closed smooth curve. The total absolute curvature is its length <math>l(\gamma)</math>.
The curve <math>\gamma</math> does not lie in an open hemisphere. If so, then there is <math>v\in\mathbb{S}^2</math> such that <math>\gamma\cdot v>0</math>, so <math>\textstyle0=(\alpha(1)-\alpha(0))\cdot v=\int_0^L\gamma(t)\cdot v\,\mathrm{d}t>0</math>, a contradiction. This also shows that if <math>\gamma</math> lies in a closed hemisphere, then <math>\gamma\cdot v\equiv0</math>, so <math>\alpha</math> is a plane curve.
Consider a point <math>\gamma(T)</math> such that curves <math>\gamma([0,T])</math> and <math>\gamma([T,L])</math> have the same length. By rotating the sphere, we may assume <math>\gamma(0)</math> and <math>\gamma(T)</math> are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves <math>\gamma([0,T])</math> and <math>\gamma([T,L])</math> intersects with the equator at some point <math>p</math>. We denote this curve by <math>\gamma_0</math>. Then <math>l(\gamma)=2l(\gamma_0)</math>.
We reflect <math>\gamma_0</math> across the plane through <math>\gamma(0)</math>, <math>\gamma(T)</math>, and the north pole, forming a closed curve <math>\gamma_1</math> containing antipodal points <math>\pm p</math>, with length <math>l(\gamma_1)=2l(\gamma_0)</math>. A curve connecting <math>\pm p</math> has length at least <math>\pi</math>, which is the length of the great semicircle between <math>\pm p</math>. So <math>l(\gamma_1)\ge2\pi</math>, and if equality holds then <math>\gamma_0</math> does not cross the equator.
Therefore, <math>l(\gamma)=2l(\gamma_0)=l(\gamma_1)\ge2\pi</math>, and if equality holds then <math>\gamma</math> lies in a closed hemisphere, and thus <math>\alpha</math> is a plane curve.
References
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal; see especially equation 13, page 49
- Шаблон:Cite book
- Шаблон:Cite book
