Fenchel's theorem
| Type | Theorem |
|---|---|
| Field | Differential geometry |
| Statement | A smooth closed space curve has total absolute curvature [math]\displaystyle{ \ge 2\pi }[/math], with equality if and only if it is a convex plane curve |
| First stated by | Werner Fenchel |
| First proof in | 1929 |
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]\displaystyle{ 2\pi }[/math]. Equivalently, the average curvature is at least [math]\displaystyle{ 2 \pi/L }[/math], where [math]\displaystyle{ L }[/math] is the length of the curve. The only curves of this type whose total absolute curvature equals [math]\displaystyle{ 2\pi }[/math] and whose average curvature equals [math]\displaystyle{ 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 4π.
Proof
Given a closed smooth curve [math]\displaystyle{ \alpha:[0,L]\to\mathbb{R}^3 }[/math] with unit speed, the velocity [math]\displaystyle{ \gamma=\dot\alpha:[0,L]\to\mathbb{S}^2 }[/math] is also a closed smooth curve. The total absolute curvature is its length [math]\displaystyle{ l(\gamma) }[/math].
The curve [math]\displaystyle{ \gamma }[/math] does not lie in an open hemisphere. If so, then there is [math]\displaystyle{ v\in\mathbb{S}^2 }[/math] such that [math]\displaystyle{ \gamma\cdot v\gt 0 }[/math], so [math]\displaystyle{ \textstyle0=(\alpha(1)-\alpha(0))\cdot v=\int_0^L\gamma(t)\cdot v\,\mathrm{d}t\gt 0 }[/math], a contradiction. This also shows that if [math]\displaystyle{ \gamma }[/math] lies in a closed hemisphere, then [math]\displaystyle{ \gamma\cdot v\equiv0 }[/math], so [math]\displaystyle{ \alpha }[/math] is a plane curve.
Consider a point [math]\displaystyle{ \gamma(T) }[/math] such that curves [math]\displaystyle{ \gamma([0,T]) }[/math] and [math]\displaystyle{ \gamma([T,L]) }[/math] have the same length. By rotating the sphere, we may assume [math]\displaystyle{ \gamma(0) }[/math] and [math]\displaystyle{ \gamma(T) }[/math] are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves [math]\displaystyle{ \gamma([0,T]) }[/math] and [math]\displaystyle{ \gamma([T,L]) }[/math] intersects with the equator at some point [math]\displaystyle{ p }[/math]. We denote this curve by [math]\displaystyle{ \gamma_0 }[/math]. Then [math]\displaystyle{ l(\gamma)=2l(\gamma_0) }[/math].
We reflect [math]\displaystyle{ \gamma_0 }[/math] across the plane through [math]\displaystyle{ \gamma(0) }[/math], [math]\displaystyle{ \gamma(T) }[/math], and the north pole, forming a closed curve [math]\displaystyle{ \gamma_1 }[/math] containing antipodal points [math]\displaystyle{ \pm p }[/math], with length [math]\displaystyle{ l(\gamma_1)=2l(\gamma_0) }[/math]. A curve connecting [math]\displaystyle{ \pm p }[/math] has length at least [math]\displaystyle{ \pi }[/math], which is the length of the great semicircle between [math]\displaystyle{ \pm p }[/math]. So [math]\displaystyle{ l(\gamma_1)\ge2\pi }[/math], and if equality holds then [math]\displaystyle{ \gamma_0 }[/math] does not cross the equator.
Therefore, [math]\displaystyle{ l(\gamma)=2l(\gamma_0)=l(\gamma_1)\ge2\pi }[/math], and if equality holds then [math]\displaystyle{ \gamma }[/math] lies in a closed hemisphere, and thus [math]\displaystyle{ \alpha }[/math] is a plane curve.
References
- do Carmo, Manfredo P. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, NY: Dover Publications, Inc.. ISBN 978-0-486-80699-0.
- "Über Krümmung und Windung geschlossener Raumkurven" (in de). Mathematische Annalen 101 (1): 238–252. 1929. doi:10.1007/bf01454836. https://eudml.org/doc/159330.
- "On the differential geometry of closed space curves". Bulletin of the American Mathematical Society 57 (1): 44–54. 1951. doi:10.1090/S0002-9904-1951-09440-9.; see especially equation 13, page 49
- O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdam: Academic Press. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4.
- Spivak, Michael (1999). A comprehensive introduction to differential geometry. Vol. III (Third edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc.. ISBN 0-914098-72-1.
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