Asymptotic line

A curve $ \Gamma $ on a regular surface $ F $ such that the normal curvature along $ \Gamma $ is zero. An asymptotic line is given by the differential equation:

$$ \textrm{ II } = L du ^ {2} +2M du dv + N dv ^ {2} = 0 , $$

where II is the second fundamental form of the surface.

The osculating plane of an asymptotic line $ \Gamma $, if it exists, coincides with the tangent plane to $ F $( at the points of $ \Gamma $), and the square of the torsion of an asymptotic line is equal to the modulus of the Gaussian curvature $ K $ of the surface $ F $( the Beltrami–Enneper theorem). A straight line $ l \in F $( e.g. a generating line of a ruled surface) is always an asymptotic line. If $ \Gamma $ is a parabolic curve (e.g. a circle on a standard torus separating the domains of Gaussian curvatures of different signs), it is an asymptotic line.

A unique asymptotic line, which coincides with the rectilinear generator, passes through each point of a parabolic domain (where $ K = 0 $, but II $ \neq 0 $). Through each point of a hyperbolic domain (where $ K<0 $) there pass exactly two asymptotic lines, forming the so-called asymptotic net, which plays an important role in the study of the spatial form of a surface of negative curvature (cf. Negative curvature, surface of). For instance, on a complete surface this net is homeomorphic to the Cartesian net on the plane if

$$ \mathop{\rm grad} \left | \frac{1} {\sqrt {-K}} \right | \leq q,\ \ q = \textrm{ const } . $$

Asymptotic nets on surfaces of constant negative curvature are Chebyshev nets (cf. Chebyshev net), and the surface area of a quadrangle formed by asymptotic lines is proportional to the excess of the sum of its interior angles $ \alpha _ {i} $ over $ 2 \pi $:

$$ | K | S = 2 \pi - \alpha _ {1} - \alpha _ {2} - \alpha _ {3} - \alpha _ {4} $$

(Hazzidakis' formula).

Under a projective transformation $ \pi $ of the space, the asymptotic lines of a surface $ F $ become the asymptotic lines of the transformed surface $ \pi (F) $.

Asymptotic lines on surfaces in a three-dimensional Riemannian space are defined in a similar manner. Various generalizations of the concept of an asymptotic line on manifolds imbedded in a multi-dimensional space are known; the most frequently-used one involves the concept of the second fundamental form, which is associated with a given normal vector.

References

[1]	A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[2]	P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)

Comments

Hazzidakis' formula can be found in [a1] and [a2], p. 204.

References

[a1]	J.N. Hazzidakis, "Uber einige Eigenschaften der Flächen mit konstanten Krümmungsmasz" Crelle's J. Math. , 88 (1880) pp. 68–73
[a2]	D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950)
[a3]	C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
[a4]	M. Spivak, "A comprehensive introduction to differential geometry" , 3 , Publish or Perish (1975) pp. 1–5
[a5]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a6]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)