## Transcribed Text

Question
1.
This question concerns surfaces of revolution. (See Homeworks 5 and 6 for the definition
and some earlier computations)
(i)
Compute the coefficients L's for a surface of revolution, and write your answer in matrix
format.
(ii) Compute the mean curvature H and the Gaussian curvature K (as functions on U).
(iii)
Draw
an
example
of
a
surface
of
revolution
that
has
some
areas
where
K
is
positive,
and
others where K is negative, and roughly indicate which areas are which.
(iv) NO NEED TO TURN THIS ONE IN: Compute all 16 coefficient functions Rijk fo the Riemann
curvature tensor for a surface of revolution (HINT: Gauss's equations are probably an easier
approach rather than going straight by definition).
Question 2. Prove that IIp is symmetric (or equivalently, prove that L is self-adjoint).
Question 3. Prove Proposition 5.15 (ii) in the notes relating Kin and II.
Question 4. Prove that n
1
X n 2 = (K,g)n.
Question 5. Suppose that 911 = 1 and 912 = 0 everywhere in U. Prove that
a2 ( V 922 )
(ou¹)
2 +K1922=0. =
Question 6. Viewing Kn as a function Kn : S¹ R as in class, prove that
27
H = 27 1 0 Kin Y )
d0
where 0 == /
(
X
(1) , Y ) as in Euler's theorem (Thm 5.21 in the course notes, which we did not cover
in class, but is easy to read and understand). This shows that H is really an arithmetic average of
Kin over the entire unit circle, not just an average over the two principal directions.

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