Beltrami-Enneper theorem

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A theorem establishing the following property of asymptotic lines (cf. Asymptotic line) on a surface of negative curvature (E. Beltrami, 1866; A. Enneper, 1870): If the curvature of an asymptotic line at a given point is non-zero, then the square of the torsion of this line is equal to the absolute value of the curvature of the surface at this point. The theorem can also be applied to the case when the curvature of the asymptotic line at a given point is zero. The square of the torsion is replaced by the square of the rate of rotation of the tangent plane to the surface at this point during the motion along the asymptote. Asymptotic lines emanating from the same point have torsions of equal absolute values and of opposite signs.


A useful general reference is [a1].


[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 3 , Publish or Perish (1975) pp. 1–5
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Beltrami-Enneper theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article