Lecture Notes for 120 - UCLA Department of Mathematics

5.3. THE GAUSS AND MEAN CURVATURES 116
(10) Compute the Gauss curvatures of the generalized cones, cylinders, and
tangent developables. We shall show below that these are essentially the
only surfaces with vanishing Gauss curvature.
(11) Show that
@N
@u ⇥ @N
@v
@q
@u ⇥ @N
@v + @N
@u ⇥ @q
@v
and more generally that
@q
@u ⇥ @N
= K @q
@u ⇥ @q
@v
= 2H @q
@u ⇥ @q
@v
@w = @q
Lv w
@u ⇥ @q
@v
@N
@w ⇥ @q = L u @q
w
@v
@u ⇥ @q
@v
(12) Compute the first and second fundamental forms as well as the Gauss and
mean curvatures for the conoid
q (s, t) = (sx (t) ,sy(t) ,z(t))
= (0, 0,z(t)) + s (x (t) ,y(t) , 0)
when X =(x (t) ,y(t) , 0) is a unit field.
(13) Let X, Y 2 T p M be an orthonormal basis for the tangent space at p to the
surface M. ProvethatthemeanandGausscurvaturescanbecomputed
as follows:
H = 1 (II (X, X)+II(Y,Y )) ,
2
K = II(X, X)II(Y,Y ) (II (X, Y )) 2
(14) Show that Enneper’s surface
✓
1
q (u, v) = u
3 u3 + uv 2 ,v
is minimal.
(15) Show that Scherk’s surface
e z cos x = cos y
◆
1
3 v3 + vu 2 ,u 2 v 2
is minimal.
(16) Consider a unit speed curve ↵ (s) :[0,L] ! R 3 with non-vanishing curvature
and the tube of radius R around it
q (s, )=↵ (s)+R (N ↵ cos + B ↵ sin )
Use Gauss’ formula for K to show that
apple cos
K =
R (1 appleR)
and use this to show that
and
ˆ 2⇡ ˆ L
0
ˆ 2⇡ ˆ L
0
0
0
K p det [I]dsd =0
|K| p det [I]dsd =4
ˆ b
a
appleds