Lecture Notes for 120 - UCLA Department of Mathematics

6.4. THE GAUSS AND CODAZZI EQUATIONS 149
tangent fields are principal directions or said differently the coordinate
curves are lines of curvature:
✓ ◆ @q @q
L = apple 1
@u @u ,
✓ ◆ @q @q
L = apple 2
@v @v .
Show that the Codazzi equations can be written as
@apple 1
@v
@apple 2
@u
= 1 2 (apple 2 apple 1 ) @ ln g uu
,
@v
= 1 2 (apple 1 apple 2 ) @ ln g vv
@u .
(5) (Hilbert) The goal is to show that if there is a point p on a surface with
positive Gauss curvature, where apple 1 has a maximum and apple 2 aminimum,
then the surface has constant principal curvatures. We assume otherwise,
in particular apple 1 (p) >apple 2 (p) , and construct a coordinate system where
the coordinate curves are lines of curvature. At p we have
@apple 1
= @apple 1
@u @v =0, @ 2 apple 1
apple 0,
@v2 @apple 2
= @apple 2
@u @v =0, @ 2 apple 2
@u 2 0.
Using the Codazzi equations from the previous exercise show that at p
@ ln g uu
@v
=0= @ ln g vv
@u
and after differentiation also at p that
@ 2 ln g uu
@v 2 0,
@ 2 ln g vv
@u 2 0
Next show that at p
K = 1 ✓ 1 @ 2 ln g uu
2 g vv @v 2 + 1 @ 2 ◆
ln g vv
g uu @u 2 apple 0
This contradicts our assumption about the Gauss curvature.
(6) Using the developments in the previous exercise show that a surface with
constant principal curvatures must be part of a plane, sphere, or right circular
cylinder. Note that the two former cases happen when the principal
curvatures are equal.
(7) (Beltrami) Assume that q (u, v) is a parametrized surface with the property
that all geodesics are lines in the domain:
(a) Show that
au + bv + c =0
v
uu =
u
vv =0,
u
uu = 2 v uv,
v
vv = 2 u uv