Transcribed Text
Question 1. Show that a unit speed meridian on a surface of revolution (see Homework 4) is always
a
geodesic (HINT: Use Prop 4.17 in the notes). Use similar techniques to determine when a circle
of latitude is a geodesic. Phrase your answer in terms of the tangent vector
X
1 along the circle of
latitude.
Question 2. Let M be a surface, and let II be a plane that intersects M symmetrically (that is to
say, the two sides of M on either side of II are mirror images of each other). Show that the curve
== M n II must then be a geodesic (once it is parametrized by unit speed). (HINT:
What
does
the symmetry condition tell you about n along 7? Give a visual of this situation, but no need
for an analytic proof)
Question 3. Show that any geodesic on the unit sphere (with full domain R) is a great circle
(see Example 4.19). HINT: There may be several possible approaches, but one would be to use
Homework 7 to help explain why
//

. The general solution of this 2nd order ODE has two
constants (constant vectors) of integration, and you should be able to use the fact that both
and
are unit vectors to help with these.
Question 4. Suppose that (U) is a simple surface having 911 = 1 and 912 = 0 everywhere. Prove
that in this case, u¹curves (curves
o
Y U where Yu has constant u2 value) are geodesics.
Question 5. Let (t) be a geodesic (with domain Rt for simplicity) that is NOT parametrized by
arc length. Let S denote arc length and P : Rt
Rs be the arc length reparametrization function
S = (t), with inverse 0 : Rs
Rt so that t = 0(s).
Prove that the downstairs curve
Y
U (t) satisfies
d²~k
i
d~² dt dyj dt
dyk
dy
2 d20
+
dt²
dt
dt
ds²
ij
for k = 1,2.
Question 6. Following up on Example 5.3 in the notes, suppose again that M = S² and p =
be the east pole, but this time define
f
S2
R
U
x
y
x2 + 3y + 4z
Z
If X =
a
is any tangent vector at p (here a, b are two constants), then compute
f.
b
Question 7. What (linear) map is the Weingarten map at any point of the unit sphere?
There
should be a oneline explanation (you will be asked for a computational proof later).
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