Page 338 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 26: Lines of Curvature

26.6 Summary Notes

A curve  on a parametric surface X is called a line of curvature if   is a principal direction.

The following proposition, due to Rodriguez, characterizes lines of curvature as those
curves whose tangents are parallel to the tangent of their spherical image under the Gauss
map.

Let  be a curve on a parametric surface X with unit normal N, and let   N  be its


spherical image under the Gauss map. Then  is a line of curvature if and only if

    0.

Let X : U   be a parametric surface, and let Y and Y be linearly independent vector
3
 1 2
fields. The following statements are equivalent:
Any point u  U has a neighborhood U and a reparametrization : V  U such

 0 0 0 0


that if X  X  then X  Y  .


i
i

Y ,Y  0.
 1 2
Let X : U   be a parametric surface, and let Y and Y be linearly independent vector
3
 1 2
fields. Then for any point u  U there is a neighborhood of u and a reparametrization
0
0


X  X  such that X  f Y  for some functions f . i


i
i
i
Let X : U   be a parametric surface, and let u be a hyperbolic point. Then there is
3
 0


neighborhood U of u and a diffeomorphism  : U  U such that X  X  is

0
0
0
0
parametrized by asymptotic lines.
26.7 Keywords
Line of curvature: A curve  on a parametric surface X is called a line of curvature if   is a
principal direction.
Bernstein’s Theorem: Let X be a minimal surface which is a graph over an entire plane. Then X is
a plane.
26.8 Self Assessment
1. A curve  on a parametric surface X is called a ................. if   is a principal direction.
2. Let  be a curve on a parametric surface X with unit normal N, and let   N  be its

spherical image under the Gauss map. Then  is a line of curvature if and only if .................
3. Let X : U   be a ................., and let Y and Y be linearly independent vector fields.
3
1
2
4. A curve  on a parametric surface X is called an ................. if it has zero normal curvature,
, 
i.e.,  k    0.
 
5. A parametric surface X is minimal if it has vanishing mean .................

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