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But we also know that
g
(
)
()
vv
S
c
s
for all tangent vectors
v
of
S
2
at g(s) and some constant c because every point of the
sphere
S
2
is an umbilical point. In our case of a unit sphere, c = 1. These two equa-
tions imply that k = 1 and t = 0. Claim 1 is proved.
Let
p
0
=g(s
0
) be a point on the curve.
Claim 2.
The curve g(s) lies in the plane
X
through the origin with basis g¢(s
0
)
and g≤(s
0
) (=
p
0
).
To prove Claim 2, note that g(s) is a unit normal to
S
2
at g(s), so that
(
)
¢=
(
)
¢= ≤¥
(
)
=
gg
¢¥ ≤
g g g
¢¥
k
k
gg
+ ¢¥
k
g
¢+ ¢
k
g
0
.
Therefore, g¢(s) ¥g≤(s) is a constant vector. If
b
=g¢(s
0
) ¥g≤(s
0
), then
(
g
()
-
)
∑=
pb
0,
0
which shows that g(s) lies in plane through
p
0
that has normal vector
b
. Given the
definition of
b
, this proves Claim 2 and also shows that the curve lies in the great
circle that is the intersection of the plane
X
and
S
2
.
To finish the proof of the converse, pick an orthonormal basis
u
1
and
u
2
for the
plane
X
, express g(s) in the form
()
=
(
()
)
(
()
)
g s
cos
u s
u
+
sin
u s
u
,
1
2
and show that u(s) has the right form with an argument like in Example 9.10.7.
9.10.9. Example.
We find some geodesics on surfaces of revolution. Assume that
we are given a surface of revolution in
R
3
and that the curve being revolved does not
intersect the axis about which it is being revolved (something that typically would
cause singularities in the surface). What we want show is that all the meridians of
this surface are geodesics and a circle of latitude is a geodesics if and only if all the
tangent planes at points of the circle are parallel to the axis of revolution.
Solution.
We shall only consider the case where a regular curve h(t) = (x(t),y(t)),
t Π[a,b], in the plane is revolved about the x-axis and the surface has a parameteri-
zation of the form
()
=
(
() ()
()
)
F t
,
q
x t
,
y t
cos
q
,
y t
sin
q
.
See Figure 9.27. We assume that F(t,q) is a regular parameterization. The proof will
rely on Theorem 9.10.2 and Theorem 9.10.3(1). Now
()
=¢
()
¢
()
(
¢
()
)
F
F
t
t
,
q
x t
,
y t
cos
q
,
y t
sin
q
()
=-
()
(
()
)
t
,
q
0
,
y t
sin
q
,
y t
cos
q
q
