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Fenchel's theorem

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Fenchel's theorem
TypeTheorem
FieldDifferential geometry
StatementA smooth closed space curve has total absolute curvature , with equality if and only if it is a convex plane curve
First stated byWerner Fenchel
First proof in1929

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 . Equivalently, the average curvature is at least , where is the length of the curve. The only curves of this type whose total absolute curvature equals and whose average curvature equals 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 .

Proof

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Given a closed smooth curve with unit speed, the velocity is also a closed smooth curve (called tangent indicatrix). The total absolute curvature is its length .

The curve does not lie in an open hemisphere. If so, then there is such that , so , a contradiction. This also shows that if lies in a closed hemisphere, then , so is a plane curve.

Consider a point such that curves and have the same length. By rotating the sphere, we may assume and are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves and intersects with the equator at some point . We denote this curve by . Then .

We reflect across the plane through , , and the north pole, forming a closed curve containing antipodal points , with length . A curve connecting has length at least , which is the length of the great semicircle between . So , and if equality holds then does not cross the equator.

Therefore, , and if equality holds then lies in a closed hemisphere, and thus is a plane curve.

References

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  • 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. MR 3837152. Zbl 1352.53002.
  • Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Mathematische Annalen (in German). 101 (1): 238–252. doi:10.1007/bf01454836. JFM 55.0394.06. MR 1512528. S2CID 119908321.
  • Fenchel, Werner (1951). "On the differential geometry of closed space curves". Bulletin of the American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. MR 0040040. Zbl 0042.40006.; 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. MR 2351345. Zbl 1208.53003.
  • 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. MR 0532832. Zbl 1213.53001.
  • Thomas F. Banchoff. "Differential Geometry". Brown University Math Department. Retrieved 2024-05-26. Fenchel's Theorem Theorem: The total curvature of a regular closed space curve C is greater than or equal to 2π.
  • Thomas F. Banchoff. "2. Curvature and Fenchel's Theorem". Brown University Math Department. Retrieved 2024-05-26.