Beltrami-Enneper theorem

The set of Beltrami Enneper (by Eugenio Beltrami and Alfred Enneper ) is a result of the differential geometry of the surfaces.

statement

The square of the torsion of an asymptote line is equal to the negative Gaussian curvature of the surface in which the curve moves, provided that the curvature of the curve itself does not vanish. A curve on a surface is called an asymptote line if the second fundamental form of the surface disappears along the curve. In particular, the Gaussian curvature is non-positive at any point on an asymptote line.

Application example

From Beltrami-Enneper's theorem it follows: If there is a regular surface that contains a straight line ( parameterization according to the arc length) and a tangential, orthogonal unit vector field along , then the curvature of in is equal ${\displaystyle X}$ ${\displaystyle L=\{\gamma (t)\}}$ ${\displaystyle \gamma }$ ${\displaystyle \lambda (t)}$ ${\displaystyle X}$ ${\displaystyle L}$ ${\displaystyle \gamma (t)}$ ${\displaystyle X}$ ${\displaystyle \gamma (t)}$

${\displaystyle -\left|{\frac {d\lambda (t)}{dt}}\right|^{2}}$

Single-shell hyperboloid

Be the single-shell hyperboloid and ${\displaystyle X=\{x^{2}+y^{2}-z^{2}=1\}}$

${\displaystyle \gamma (t)=\left(1,{\frac {t}{\sqrt {2}}},{\frac {t}{\sqrt {2}}}\right),\lambda (t)={\frac {1}{\sqrt {1+t^{2}}}}\cdot \left(-t,{\frac {1}{\sqrt {2}}},-{\frac {1}{\sqrt {2}}}\right)\,.}$

Then

${\displaystyle {\frac {d\lambda (t)}{dt}}={\frac {1}{(1+t^{2})^{3/2}}}\cdot \left(-1,-{\frac {t}{\sqrt {2}}},{\frac {t}{\sqrt {2}}}\right)}$

and thus

${\displaystyle -\left|{\frac {d\lambda (t)}{dt}}\right|^{2}={\frac {-1}{(1+t^{2})^{2}}}=K\,.}$

Individual evidence

1. ^ Eugenio Beltrami: Dimostrazione di due formole del Sig. Bonnet. In: Giornale di Matematiche. 4, 1866, ZDB -ID 281094-3 , pp. 123-127 (also in: Eugenio Beltrami: Opere matematiche. Volume 1. Hoepli, Milan 1902, pp. 297-301), online .
2. ^ Alfred Enneper: About asymptotic lines. In: News from the Georg August University and the Königl. Society of Sciences at Göttingen. 12, 1870, ZDB -ID 502554-0 , pp. 493-510, (result is formulated on p. 499), online
3. ↑ W. Blaschke , K. Leichtweiß : Elementare Differentialgeometrie (= lectures on differential geometry. 1 = The basic teachings of the mathematical sciences in individual representations. 1). 5th completely revised edition. Springer-Verlag, Berlin a. a. 1973, ISBN 3-540-05889-3 , § 56, pp. 133f.
4. ↑ Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds. 5th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1233-9 , sentence 3.19, p. 57.
5. ↑ Victor Andreevich Toponogov : Differential geometry of curves and surfaces. A concise guide. Birkhäuser, Boston a. a. 2006, ISBN 0-8176-4384-2 , theorem 2.7.6.