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be the standard parameterization of
S
. Show that
(
)
xxy xy
yx
¢≤¢-¢≤
(a)
K
=
(
)
2
2
¢
+
y
¢
(b)
If g(t) is the arc-length parameterization, then K =-y≤/y.
(c)
Assume that x(t) = t. Show that every point of
S
is a parabolic point if and only if
S
is a cylinder (y(t) = c) or a cone (y(t) = at + b, a π 0).
9.9.5.
Show that a definition of the second fundamental form based on Equation (9.47) is inde-
pendent of the choice of local coordinates.
9.9.6.
Let
p
be a point of a surface
S
. Let L, M, and N be the coefficients of the second fun-
damental form of
S
. Show that the point
p
is
(1) elliptic if LN - M
2
> 0,
(2) hyperbolic if LN - M
2
< 0,
(3) parabolic if LN - M
2
= 0 but not all of the L, M, and N are zero, and
(4) planar or flat if L = M = N = 0.
These characterizations of elliptic, hyperbolic, parabolic, and planar or flat are some-
times used as the definition.
9.9.7.
Find the Gauss and mean curvatures of the surface F(u,v) = (u + v, u - v,uv) at (1,1).
Section 9.10
9.10.1.
Prove Theorem 9.10.1(3). See [Stok69].
Show that the curve g(t) = (t,0,t
2
) is a geodesic in the surface F(u,v) = (u,v,u
2
- v
2
).
9.10.2.
Let
S
be a surface in
R
3
.
Definition.
Let
X
be a plane in
R
3
and let r be the reflection about
X
. We say that
X
is
a
plane of symmetry
for
S
if for all points
p
in
S
the reflected point r(
p
) belongs to
S
.
Prove that if
X
is a plane of symmetry for
S
, then the intersection of
X
and
S
defines
a geodesic curve. Use this fact to find some geodesics on the surface defined by x
2
9.10.3.
+
y
2
- z
2
= 1.
9.10.4.
Prove Theorem 9.10.15.
9.10.5.
Prove Theorem 9.10.20.
Section 9.12
9.12.1.
Let g(s) be a space curve parameterized by arc-length. Let (T(s),N(s),B(s)) be its
moving trihedron. Consider the canal surface
S
with constant radius function and
parameterization
(
)
=
()
+
(
)
()
+
(
)
()
F s
,
qg
s
r
cos
q
N s
r
sin
q
B s
.
Let k be the curvature of g and K the Gauss curvature of
S
. Prove that
(a)
g = r
2
(1 - kr cos q)
2
=-
k
cos q
(b)
K
g
