MATH 434
Fall 2016
Homework 1, due on Wednesday August 31
Problem 1. Let z = 2 − i and z 0 = 3 + 4i. Write the product zz 0 and the quotient
z
in the form a + ib, with a, b ∈ R.
z0
Problem 2. Let z ∈ C be a complex number, and let z̄ be its conjugate. Show
that z is a real number if and only if z = z̄. Namely:
a. First show that, if z is a real number, then z = z̄.
b. Then show that, if z = z̄, then z is a real number.
Problem 3. Find r and θ so that i − 1 = reiθ . Hint: First plot i − 1 in the complex
plane, and use polar coordinates.
Problem 4. Let ϕ: C → C be the rotation of angle θ around the point z0 ∈ C.
Express ϕ(z) in terms of z, z0 and eiθ . (Remember that we considered the case
z0 = 0 in class.)
Problem 5. The map ψ : C → C defined by ψ(z) = −z̄ is a relatively simple
transformation of the plane. What is it? (Namely describe it with words, such as
“the rotation of angle π7 around the point 2 − i”; of course, this is not the answer.)
1
MATH 434
Fall 2016
Homework 2, due on Friday September 9
Problem 1. Let (X, d) be a metric space.
0
0
a. Show that d(P,Q) − d(P, Q0 ) 6 d(Q,
Q ) for 0every P , Q, Q ∈ X.0
0
b. Conclude that d(P, Q) − d(P, Q ) 6 d(Q, Q ) for every P , Q, Q ∈ X.
Problem 2. Let X be the plane R2 , and let d1 , d2 , d3 : X × X → R be defined by
p
d1 (x, y), (x0 , y 0 ) = (x − x0 )2 + (y − y 0 )2
d2 (x, y), (x0 , y 0 ) = |x − x0 | + |y − y 0 |
d3 (x, y), (x0 , y 0 ) = max{|x − x0 |, |y − y 0 |}.
In particular, d1 is the usual euclidean distance deuc , and we proved in class that
(X, d1 ) is a metric space.
a. Show that (X, d2 ) is a metric space.
b. Show that (X, d3 ) is a metric space.
Problem 3. In a metric space (X, d), the ball of radius r centered at the point P
is the set
Bd (P, r) = {Q ∈ X; d(P, Q) < r}
consisting of all points Q in X such that d(P, Q) < r. For the metric spaces (X, d1 ),
(X, d2 ) and (X, d3 ) of Problem 2 and for the point P0 = (0, 0) in X = R2 , draw the
balls Bd1 (P0 , 1), Bd2 (P0 , 1) and Bd3 (P0 , 1).
MATH 434
Fall 2016
Homework 3, due on Wednesday September 14
Problem 1. In the hyperbolic plane H2 , consider the two points P = i and
Q = 4 + i. For u > 0, let Pu = ui, let Qu = 4 + ui, and let γu be the curve
going from P to Q that is made up of the vertical line segment [P, Pu ], followed by
the horizontal line segment [Pu , Qu ], and finally followed by the vertical segment
[Qu , Q].
a.
b.
c.
d.
Draw a picture of γu .
Compute the hyperbolic length `hyp (γu ).
For which value of u is `hyp (γu ) minimum? (Hint: Remember calculus?)
Use Part c to show that dhyp (P, Q) 6 2 ln 2 + 2.
Problem 2. Let ϕ: H2 → H2 be the map defined by the property that ϕ(x, y) =
(−x, y). (Namely, ϕ is the euclidean reflection across the y–axis.)
a. Show that, if γ is a curve in H2 and if γ1 is the image of γ under ϕ, then
`hyp (γ1 ) = `hyp (γ).
b. Use Part a to show that ϕ is an isometry from (H2 , dhyp ) to itself.
MATH 434
Fall 2016
Homework 4, due on Wednesday September 21
Problem 1. Given four numbers a, b, c, d ∈ C with ad − bc = 1 consider the map
az + b
ϕ(z) =
cz + d
defined for any complex number z different from − dc .
a. Given a similar map
a0 z + b0
with a0 , b0 , c0 , d0 ∈ C and a0 d0 − b0 c0 = 1,
ϕ0 (z) = 0
c z + d0
compute the composition ϕ ◦ ϕ0 and show that there exists a00 , b00 , c00 , d00 ∈ C
with a00 d00 − b00 c00 = 1 such that
a00 z + b00
ϕ ◦ ϕ0 (z) = 00
c z + d00
for every z where it is defined.
b. If
dz − b
ψ(z) =
,
−cz + a
compute ϕ ψ(z) and ψ ϕ(z) .
Remark. (No credit) If you remember from linear algebra how to multiply matrices, you may notice that
00
0
a
b00
a b
a b0
=
.
c d
c0 d0
c00 d00
This is not a coincidence. (Do not write anything. This is just intended to whet
your appetite for more math.)
Problem 2. Let
az + b
with a, b, c, d ∈ R and ad − bc = 1
cz + d
as in Problem 1, and suppose in addition that a 6= 0. Note that a, b, c, d are now
real numbers. Set
b
1
ϕ1 (z) = z +
ϕ2 (z) =
a
z
1
c
ϕ3 (z) = 2 z
ϕ4 (z) = z + .
a
a
ϕ(z) =
a. Which ones of ϕ1 , ϕ2 , ϕ3 , ϕ4 are horizontal translations, homotheties or
inversions?
b. Show that
ϕ = ϕ2 ◦ ϕ4 ◦ ϕ3 ◦ ϕ2 ◦ ϕ1 .
c. Show that ϕ defines an isometry of the hyperbolic plane (H2 , dhyp ). (Hint:
Part b.)
MATH 434
Fall 2016
Homework 5, due on Wednesday September 29
Problem 1. Inspired by what we did in class for isometries of the hyperbolic plane
(H2 , dhyp ), the goal of this problem is to describe all isometries of the euclidean plane
(R2 , deuc ). More precisely, we will rigorously prove that all isometries of (R2 , deuc )
are the ones we saw in class a few weeks ago, and the proof will be cut into several
steps. In particular, each question usually relies on the previous ones.
a. Consider the two points P1 = (0, 0) and P2 = (1, 0). Show that, for any
two positive numbers d1 and d2 , there exists exactly zero, one or two points
P = (x, y) such that deuc (P, P1 ) = d1 and deuc (P, P2 ) = d2 .
When they are two such points, show that they are related to each other
by reflection across the x–axis.
(Hint: Express deuc (P, P1 ) and deuc (P, P2 ) in terms of x and y, and solve.)
b. Consider in addition the point P3 = (0, 1). Show that if the two points
P = (x, y) and P 0 = (x0 , y 0 ) are such that deuc (P, P1 ) = deuc (P 0 , P1 ),
deuc (P, P2 ) = deuc (P 0 , P2 ) and deuc (P, P3 ) = deuc (P 0 , P3 ), then necessarily
P = P 0.
c. Let ϕ : R2 → R2 be an isometry of (R2 , deuc ) such that ϕ(P1 ) = P1 , ϕ(P2 ) =
P2 and ϕ(P3 ) = P3 . Show that ϕ(P ) = P for every P ∈ R2 .
d. Let ϕ : R2 → R2 be an isometry of (R2 , deuc ) such that ϕ(P1 ) = P1 and
ϕ(P2 ) = P2 . Show that ϕ is, either the identity map defined by ϕ(x, y) =
(x, y), or the reflection ϕ(x, y) = (x, −y) across the x–axis.
e. Let ϕ : R2 → R2 be an isometry of (R2 , deuc ). Show that there exists a
translation ψ1 that sends ϕ(P1 ) to P1 . Show
exists a rotation ψ2
that there
around the point P1 = (0, 0) such that ψ2 ψ1 ϕ(P2 ) = P2 . (You may need
to use the fact that ψ1 and ϕ are isometries.) Show that, for the composition
ψ = ψ2 ◦ ψ1 , there exists z1 ∈ C and an angle θ1 ∈ R such that, in complex
coordinates, ψ(z) = eiθ1 z + z1 .
f. For ϕ and ψ as in Part e, show that the composition ψ ◦ ϕ is an isometry of
(R2 , deuc ) that sends P1 to P1 , and sends P2 to P2 .
g. Combine Parts d, e and f (and a short computation) to show that, for every
isometry ϕ of the euclidean plane (R2 , deuc ), there exists z0 ∈ C and θ ∈ R
such that, either ϕ(z) = eiθ z + z0 for every z ∈ C, or ϕ(z) = e2iθ z̄ + z0 for
every z ∈ C.
Problem 2. Problem 1 is long enough. There is no Problem 2. ^
¨
Math 434
Practice Midterm
Fall 2016
The actual exam will have one fewer problem. Do not turn this in.
Problem 1. Consider the two points P = (−2, 2) and Q = (2, 2) in the hyperbolic
plane (H2 , dhyp ).
a. Compute the hyperbolic length `hyp [P, Q] of the line segment [P, Q].
b. What is the shortest curve going from P to Q (where “shortest” means
“shortest for the hyperbolic arc length `hyp ”)?
c. Give a parametrization of this shortest curve from P to Q.
d. Compute the hyperbolic distance dhyp (P, Q).
Problem 2. On a set X, define
(
d(P, Q) =
0
1
if P = Q
if P =
6 Q
for every two points P , Q ∈ X.
Show that (X, d) is a metric space. (Remember that there are four conditions
to check.)
Problem 3. Let ϕ : X → X be an isometry of the metric space (X, d), such that
ϕ(P0 ) = P0 for some point P0 ∈ X. Show that ϕ sends each P ∈ X to a point ϕ(P )
that is at the same distance from P0 as P , namely such that d ϕ(P ), P0 = d(P, P0 ).
Problem 4.
a. Show that
(cos θ + i sin θ)5 = cos 5θ + i sin 5θ
for every θ ∈ R. Hint: eiθ .
b. Use Part a to show that
cos 5θ = cos5 θ − 10 cos3 θ sin2 θ + 5 cos θ sin4 θ
for every θ ∈ R.
Problem 5. Let ϕ: H2 → H2 be the isometry of (H2 , dhyp ) defined by
az + b
with a, b, c, d ∈ R and ad − bc = 1.
cz + d
Suppose in addition that |a + d| > 2 and c 6= 0.
ϕ(z) =
a. Show that there exists exactly two points x ∈ R such that ϕ(x) = x. Hint:
quadratic formula.
b. Use Part a to show that there is a unique complete geodesic g in H2 such
that ϕ(g) = g.
Math 434
Actual midterm
Fall 2016
The percentages denote the percentage of points assigned to each problem/subproblem.
Problem 1. (Total: 20%)
a. (10%) Give the x– and y–coordinates of the point corresponding to the
π
complex number z = 2e−i 4 .
√
b. (10%) Find r and θ such that 1 + i 3 = reiθ .
√
Problem 2. (Total: 30%) Consider the points P = (1, 3) and Q = (0, 2) in the
hyperbolic plane (H2 , dhyp ).
a. (6%) What are the polar coordinates of P and Q?
b. (8%) What is the shortest curve going from P to Q (where “shortest” means
“shortest for the hyperbolic length `hyp ”)?
c. (8%) Give a parametrization of this shortest curve from P to Q.
d. (8%) Express the hyperbolic distance dhyp (P, Q) as an integral of explicit
functions, but do not try to compute this
integral. (Namely, leave your
Z √13
t3 + cos 5t
ln p
answer as something like dhyp (P, Q) =
dt.)
π
sin3 t + 5
7
Problem 3. (Total: 25%) Let f : R → R be any positive continuous function
defined on the real line R (namely, f (x) > 0 for every x ∈ R). Define a function
d : R × R → R of two variables by


sup{f (z); x 6 z 6 y} if x < y
d(x, y) = sup{f (z); y 6 z 6 x} if y < x


0
if x = y.
Namely, d(x, y) is the supremum of the values taken by f between x and y.
Show that (R, d) is a metric space. (It may be useful to remember the Extreme
Value Theorem from calculus, which says that the function f achieves its maximum
over each closed interval [a, b]; namely, for every closed interval [a, b], there exists
c ∈ [a, b] such that f (c) = sup{f (x); x ∈ [a, b]}.)
Problem 4. (Total: 25%) Let ψ: H2 → H2 be the (antilinear fractional) isometry
of (H2 , dhyp ) defined by
cz̄ + d
with a, b, c, d ∈ R and ad − bc = 1.
az̄ + b
Suppose in addition that b + c = 0 and a 6= 0.
ψ(z) =
Show that the set of points z ∈ H2 such that ψ(z) = z is a complete geodesic,
namely a semi-circle centered on the x–axis. (You may find it convenient to switch
to cartesian coordinates after the preliminary steps of the computation.)
MATH 434
Fall 2016
Homework 6, due on Wednesday October 19
Recall from several weeks ago that, in a metric space (X, d), the ball of radius
r centered at P ∈ X is Bd (P, r) = {Q ∈ X; d(P, Q) < r}. The three problems are
devoted to these balls, in various spaces.
Note that the assignment continues on the next page.
Problem 1. We first consider the disk model (B2 , dB2 ). Let 0 be the center of the
disk B2 .
a. For a point P ∈ B2 , express the B2 –length `B2 [0, P ] of the line segment
[0, P ] in terms of the euclidean distance D = deuc (0, P ). In the, I know, very
unlikely event that you forgot about partial fractions I remind you that
Z
Z
Z
1 + x
dx
dx
dx
1
1
1
=
+
=
ln
2
2
1 − x + C.
1 − x2
1+x 2
1−x
b. For 0 and P as in Part a, what is the shortest curve from 0 to P ? What is
its length? What is the distance dB2 (0, P )?
c. Show that the ball BdB2 (0, r) in B2 coincides with the euclidean open disk
of radius tanh 2r =
r
r
e 2 −e− 2
r
r
e 2 +e− 2
centered at 0.
Problem 2. We now consider the hyperbolic plane (H2 , dhyp ), and the isometry
Φ : H2 → B2 from (H2 , dhyp ) to (B2 , dB2 ) defined by
−z + i
.
z+i
Also consider the linear fractional map Ψ defined by
Φ(z) =
Ψ(z) =
−iz + i
.
z+1
a. Show that Φ ◦ Ψ(z) = Ψ ◦ Φ(z) = z for every z. Conclude that Ψ sends
every point of B2 to a point of H2 , and defines an isometry from (B2 , dB2 ) to
(H2 , dhyp ).
b. Use Part a to show that, for the point i = Ψ(0), the ball Bdhyp (i, r) is the
image of the ball BdB2 (0, r) under Ψ, namely that Bdhyp (i, r) = Ψ BdB2 (0, r) .
c. Use Problem 1 and a certain property of linear fractional maps to show that
Bdhyp (i, r) is bounded by a (euclidean) circle C.
d. Show that Ψ sends the x–axis to the y–axis. Conclude that the circle C
contains the points Ψ(tanh 2r ) = e−r , Ψ(− tanh 2r ) = er , and is orthogonal
to the y–axis.
e. Show that the ball Bdhyp (i, r) is the open euclidean disk whose euclidean
center is i cosh r (not i!) and whose euclidean radius is sinh r. Just in case,
r
−r
r
−r
I remind you that cosh r = e +e
and sinh r = e −e
.
2
2
f. Show that, for every y > 0, the ball Bdhyp (iy, r) is an open euclidean disk.
What are its euclidean center and euclidean radius? (Hint: homothety.)
g. Show that, for every z = x+iy ∈ H2 , the ball Bdhyp (z, r) is an open euclidean
disk. What are its euclidean center and euclidean radius? (Hint: horizontal
translation.)
Problem 3. In the sphere S2 , let N = (0, 0, 1) be the “North Pole”. Describe each
of the balls Bdsph (N, π2 ), Bdsph (N, π), Bdsph (N, 3π
2 ) and Bdsph (N, 2π) with a picture
and a few words.
MATH 434
Fall 2016
Homework 7, due on Wednesday October 26
Problem 1. Remember that the metric space (X, d) is locally isometric to the
metric space (X 0 , d0 ) if, for every P ∈ X, there exists an isometry ϕ: Bd (P, r) →
Bd0 (P 0 , r) from a small ball Bd (P, r) centered at P in X and a small ball Bd0 (P 0 , r)
in X 0 . Also, (X, d) is locally homogeneous if, for every P , Q ∈ X, there exists an
isometry ϕ: Bd (P, r) → Bd (Q, r) from a small ball Bd (P, r) centered at P in X to
a small ball Bd (Q, r) centered at Q.
Show that, if (X, d) is locally isometric to (X 0 , d0 ) and if (X 0 , d0 ) is locally homogeneous, then (X, d) is locally homogeneous.
Problem 2. In the plane X = R2 , consider for each c ∈ R the hyperbola
Hc = {(x, y) ∈ R2 ; xy = c}.
(When c = 0, the “hyperbola” H0 is somewhat degenerate.)
a. Draw a picture of H1 , H−1 , H0 and H 21 .
b. Show that the hyperbolas Hc form a partition X̄ of X = R2 , in the sense
that every point P ∈ R2 belongs to one and only one hyperbola Hc .
c. Consider the hyperbolas Hc1 and Hc2 associated to positive numbers c1 ,
c2 > 0. Show that, for every ε > 0, there exist two points P1 ∈ Hc1 and
P2 ∈ Hc2 such that deuc (P1 , P2 ) < ε.
d. More generally, consider the hyperbolas Hc1 and Hc2 associated to arbitrary
numbers c1 , c2 ∈ R. Show that, for every ε > 0, there exist two points
P1 ∈ Hc1 and P2 ∈ Hc2 such that deuc (P1 , P2 ) < ε.
e. Let d¯euc be the quotient semi-metric on the partition X̄ defined (using discrete walks as seen in class) by the euclidean metric deuc of X = R2 . In
particular, for P ∈ R2 , let P̄ ∈ X̄ denote the hyperbola Hc that contains it.
(i) Show that d¯euc (P̄1 , P̄2 ) 6 deuc (P1 , P2 ), for every P1 ∈ Hc1 and P2 ∈
Hc2 . (Hint: Can you find a discrete walk from P̄1 to P̄2 ?)
(ii) Conclude that d¯euc (P̄1 , P̄2 ) = 0 for every P̄1 , P̄2 ∈ X̄. Hint: Part d.
(iii) Is (X̄, d¯euc ) a metric space?
MATH 434
Fall 2016
Homework 8, due on Wednesday November 3
Recall from class that a homeomorphism from the metric space (X, d) to the
metric space (X 0 , d0 ) is a bijective map ϕ : X → X 0 such that both ϕ and its
inverse ϕ−1 are continuous.
Problem 1. Let X be a regular decagon (= polygon with 10 edges and 10 vertices)
in the euclidean plane (R2 , deuc ), and let (X̄, d¯euc ) be the quotient space obtained
by gluing by euclidean translations opposite edges of the decagon X.
a. The vertices of X correspond to how many points of X̄?
b. Is the quotient space (X̄, d¯euc ) locally isometric to the euclidean plane (R2 , deuc )?
Explain.
c. Give a “proof by pictures”, like the ones we have used in class in recent weeks,
suggesting that the quotient space (X̄, d¯euc ) is homeomorphic to the surface
of genus 2 (namely the surface we already obtained by gluing opposite edges
of an octagon).
d. (No credit) If we glue opposite sides of a 2n–gon X in R2 , what do you think
the quotient space X is homeomorphic to? (Hint: do you see a pattern in
the cases n = 2, 3, 4, 5?
Problem 2. Let X be the square {(x, y) ∈ R2 ; 0 6 x 6 1, 0 6 y 6 1} in the
euclidean plane, and let (X̄, d¯euc ) be the Klein bottle obtained from X by gluing
each point (0, y) to the point (1, y), and each point (x, 0) to (1 − x, 1). Draw a
picture of X and indicate by arrows the gluing of its sides, as we have done in class.
a. Let α be the horizontal line segment {(x, y) ∈ X; y = 21 } in X. Draw a
picture of α. Show that its image ᾱ in X̄ is a closed curve, namely that its
end points are glued together.
b. For ᾱ as in Part a, let X̄ − ᾱ consists of all points P̄ ∈ X̄ that are not in ᾱ.
What is X̄ − ᾱ homeomorphic to? (Use a “proof by picture”.)
c. Let β be the vertical line segment {(x, y) ∈ X; x = 21 } in X. Draw a
picture of β. Show that its image β̄ is a closed curve in X̄. What is X̄ − β̄
homeomorphic to? (Use a “proof by picture”.)
d. Let γ consist of the two vertical line segments {(x, y) ∈ X; x = 13 or 23 } in
X. Draw a picture of γ. Show that the image γ̄ consists of a single closed
curve in X̄. What is X̄ − β̄ homeomorphic to? (Use a “proof by picture”.)
MATH 434, Fall 2016
Practice Final Exam
Problem 1. (10%) Sketch a tessellation of the euclidean plane R2 by triangles
whose angles are π2 , π3 , π6 .
Problem 2. Let X be a regular dodecagon in the hyperbolic plane (H2 , dhyp ),
with all 12 sides of equal lengths and all 12 angles equal to θ. Label the vertices of
X as V1 , V2 , . . . , V12 in this order around X, and glue the edge A1 A2 to A8 A7 , the
edge A2 A3 to A1 A12 , the edge A3 A4 to A6 A5 , the edge A4 A5 to A11 A10 , the edge
A6 A7 to A9 A8 , and the edge A9 A10 to A12 A11 . (It may help to draw arrows on the
picture below, which represents X is the disk model for symmetry.) Let (X̄, d¯hyp )
be the corresponding quotient space.
A1
A12
A11
A2
A10
A3
A9
A4
A8
A5
A7
A6
a. (8%) How many points of X̄ correspond to the vertices of X?
b. (8%) For which value of θ is the quotient space (X̄, d¯hyp ) locally isometric
to the hyperbolic plane (H2 , dhyp )?
Problem 3. (12%) Let X be a polygon in the euclidean plane R2 , and let X̄ be
the quotient space obtained by gluing edges of X together. Given two points P̄ ,
Q̄ ∈ X̄ in this quotient space, give the precise definition of a discrete walk w from
P̄ to Q̄, and of the length `d (w) of this discrete walk.
Problem 4. We want to endow the real line R with a new metric d, defined by
the property that

if x = y

0
p
1
d(x, y) = max{ q ; p, q integers, q > 0, x < q < y} if x < y


max{ 1q ; p, q integers, q > 0, y < pq < x} if x > y.
(Namely, d(x, y) is 1 over the smallest denominator of a rational number sitting
between x and y.)
a. (6%) Compute d(0, 12 ), d(0, 13 ) and d( 12 , 13 ).
b. (14%) Show that d is indeed a metric, and that (R, d) is a metric space.
Problem 5. Consider a hyperbolic isometry
az + b
ϕ(z) =
with a, b, c, d ∈ R, ad − bc = 1 and c 6= 0
cz + d
and the horizontal line
L = {x + i; x ∈ R}
defined by the equation y = 1.
a. (4%) Compute ϕ(∞).
b. (8%) Remember that we saw in class that a linear fractional map sends circle
to circle (if we consider a line plus the point ∞ as a circle of infinite radius).
Use this property to show that ϕ sends L to a C − { ac }, where C is a circle
in C that is tangent to the real line R at the point ac , and where C − { ac }
denotes the circle C from which the point ac has been removed.
c. (continuation of Problem 5) (10%) Compute the imaginary part of ϕ(x + i),
and find the maximum of this imaginary part as x ranges over all points of
R.
d. (2%) Use Part b to find the radius of the circle C.
Problem 6. (18%) Let ϕ: H2 → H2 be a hyperbolic isometry sending the point ∞
to itself. Show that, either ϕ is a horizontal translation ϕ(z) = z + x0 with x0 ∈ R,
or there exists a complete geodesic g such that ϕ(g) = g. (Possible hint: Write ϕ
cz̄+d
as ϕ(z) = az+b
cz+d or ϕ(z) = az̄+b and look for the end points of g.)