THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
November 2018
MATH3531
Differential Geometry and Topology
(1) TIME ALLOWED – 2 HOURS
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(6) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER
MAY BE USED
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
November 2018 MATH3531 Page 2
Useful Equations
For a plane curve r = (x(t), y(t)):
T =
(
x˙√
x˙2 + y˙2
,
y˙√
x˙2 + y˙2
)
N =
(
− y˙√
x˙2 + y˙2
,
x˙√
x˙2 + y˙2
)
κ˜(t) =
y¨x˙− x¨y˙
(x˙2 + y˙2)3/2
T′ = κ˜(s)N N′ = −κ˜(s)T.
For a space curve r = (x(t), y(t), z(t)):
v = ‖r˙‖ T = r˙
v
κ =
‖r˙× r¨‖
v3
B =
r˙× r¨
‖r˙× r¨‖ N = B×T τ =
[r˙, r¨,
...
r ]
‖r˙× r¨‖2TN
B
′ =
 0 κ 0−κ 0 τ
0 −τ 0
TN
B

Isometry such that f(r2) = r1 for congruent curves:
f(x) = r1(0)−Qr2(0) +Qx,
where Q = F1(0)F
T
2 (0).
For surface X(u, v):
U =
Xu ×Xv
‖Xu ×Xv‖
E = Xu ·Xu F = Xu ·Xv G = Xv ·Xv
L = U ·Xuu M = U ·Xuv N = U ·Xvv
K =
LN −M2
EG− F 2 H =
GL+ EN − 2FM
2(EG− F 2) κg =
[U, r˙, r¨]
speed3
χ = V − E + F
Reduction of combinatorial surfaces:
STEP 4 aXaY = 1 → bbX−1Y = 1
STEP 5 ZaUbV a−1Sb−1T = 1 → Zcdc−1d−1TUV S = 1
STEP 6 ccXaba−1b−1Y = 1 → eeffggX−1Y −1 = 1.
Please see over . . .
November 2018 MATH3531 Page 3
QUESTION 1
Use a separate book clearly marked Question 1
1. i) [8 marks] For each of the following, say whether the statement is true or
false and give a brief reason or show your working. You will get one mark
for a correct true/false answer, and if your true/false answer is correct
then you will get one mark for a good reason.
Begin each answer with the word “true” or “false”.
a) The curve r = ((1 + t)3/2, (3− t)3/2) for −1 ≤ t ≤ 3 has length 12.
b) If r(0) = 0, N(0) = 1
3
(2, 2,−1) and B(0) = 1
3
(2,−1, 2), then the
osculating plane of r at r(0) is 2x− y + 2z = 0.
c) If X(u, v) is a function such that Xu×Xv = (u2, v2, uv), then X(u, v)
defines a parameterised surface for all values of u and v.
d) There are exactly 5 Platonic polyhedra.
ii) [5 marks] Consider the family of straight lines in R2 that cut the x-axis
at A = (c, 0) and the y-axis at B so that the area of triangle OAB is
always s.
a) Explain why an arbitrary member of the family is given by
2sx+ c2y = 2cs ,
where c is the parameter of the family.
b) Show the envelope of this family is (one branch of) a hyperbola.
iii) [4 marks] Suppose A : (a, b) → R2 is a smooth, non-constant vector-
valued function with ‖A(s)‖ = 1 for all s ∈ (a, b). Let r1 be a curve
whose tangent is A and r2 be a curve whose principal normal is A.
Prove that r1 and r2 have the same absolute curvature.
What does this mean for the two curves?
iv) [3 marks] The curve
r(t) = (4t+ 3 cos t, 3t− 4 cos t,−5 sin t)
has curvature κ =
1
10
and torsion τ =
1
10
. (You do not need to prove
this.)
Find a curve with curvature
1
10
and torsion − 1
10
, and explain why your
answer has that curvature and torsion.
Please see over . . .
November 2018 MATH3531 Page 4
QUESTION 2
Use a separate book clearly marked Question 2
2. i) [14 marks] Let S be the catenoid: the surface given by
(coshu cos v, coshu sin v, u) u, v ∈ R.
a) Show that the tangent plane to S at the point where u = 0, v =
pi
4
is x+ y =
√
2.
b) Find the first and second fundamental forms of S.
c) Find the Gaussian curvature of S and show that the mean curvature
of S is identically zero.
d) Find the area of the catenoid between u = 1 and u = −1. Why
would you expect this to be less than the area of the cylinder passing
through those two circles?
e) An asymptotic direction on a surface is a vector v ∈ TPS for which
IIP (v,v) = 0, where II is the second fundamental form. Show that
the tangent to the curves on the catenoid with v = u + c for any
constant c is an asymptotic direction at each point.
ii) [6 marks]
a) Define what it means for two surfaces to be isometric (also called
applicable).
b) Explain why no two of the following surfaces are isometric:
A) sphere;
B) torus;
C) circular cylinder;
D) catenoid.
Please see over . . .
November 2018 MATH3531 Page 5
QUESTION 3
Use a separate book clearly marked Question 3
3. i) [5 marks] For the torus,
X = ((a+ b cosu) cos v, (a+ b cosu) sin v, b sinu) ,
show that the meridians (curves with v constant) are geodesics, and prove
that the geodesic curvature of the parallel u = k, k a constant is
κg =
sin k
a+ b cos k
.
(Use the outward normal.)
ii) [12 marks] The Gauss-Bonnet theorem is:
Suppose that C is a curvilinear polygon on surface S and has interior Ω,
and let αi be the exterior angle at vertex i of C. Then∫
C
κg ds+
∫∫
Ω
K dA+
∑
i
αi = 2pi. (1)
a) Define the term curvilinear polygon, and the terms K, dA and αi
in equation (1).
b) For the torus, given by
X = ((a+ b cosu) cos v, (a+ b cosu) sin v, b sinu) ,
verify the Gauss-Bonnet theorem over the region given by 0 ≤ u ≤ pi
and 0 ≤ v ≤ 1
4
pi.
You may assume without proof the results of part i) above and that
for the torus the first fundamental form is
ds2 = b2du2 + (a+ b cosu)2 dv2
and that K =
cosu
b(a+ b cosu)
.
iii) [3 marks] If a polygonal subdivision of a torus consists of 8 triangles
and 5 squares, with a total of 22 distinct edges then how many distinct
vertices must there be?
Please see over . . .
November 2018 MATH3531 Page 6
QUESTION 4
Use a separate book clearly marked Question 4
4. i) [4 marks] Using diagrams, show how two cross caps (a cross cap has
edge equation aab = 1) can be glued together to create a Klein bottle
(with edge equation xyxy−1 = 1).
ii) [10 marks] Reduce the combinatorial surface with the following set of
edge equations to standard form, describe the surface, compute its Euler
characteristic and state whether it is orientable or not.
{e−1bca−1 = 1, f−1d−1be = 1, a−1cfd = 1}
iii) [6 marks]
a) Define the chromatic number of a surface and state the Four
Colour Theorem.
b) Prove that for any polygonal subdivision of a surface (i.e. a map)
2E
F
≤ 6
(
1− χ
F
)
.
END OF EXAMINATION

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