The Beltrami-Enneper theorem is a theorem on the property of asymptotic lines of a surface of negative curvature.
The theorem was proved independently of each other by Eugenio Beltrami in 1866 and by Alfred Enneper in 1870 .
Formulation
If the curvature of the 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.
Remarks
- For an asymptotic curve, if an adjoining plane is defined, then it coincides with the tangent plane to the surface. Therefore, instead of the square of torsion, you need to take the square of the speed of rotation of the tangent plane at this point when offset along the asymptotic curve. This reformulation is useful when the curvature of the asymptotic line at a point is zero and therefore the contiguous plane is not defined.
Literature
- Rashevsky P.K. Course of differential geometry. M.L.: Hittl, 1950.