MATH 434 Fall 2016 Homework 1, due on Wednesday August 31

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Homework 1, due on Wednesday August 31 Problem 1. Let z = 2 i and z = 3 + 4i. Write the product zz and the quotient z z in the form a + ib, with a, b R. Problem 2. Let z C be a complex number, and let z be its conjugate. Show that z is a real number if and only if z = z. Namely: a. First show that, if z is a real number, then z = z. b. Then show that, if z = z, then z is a real number. Problem 3. Find r and θ so that i 1 = re iθ. 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 z 0 C. Express ϕ(z) in terms of z, z 0 and e iθ. (Remember that we considered the case z 0 = 0 in class.) Problem 5. The map ψ : C C defined by ψ(z) = z is a relatively simple transformation of the plane. What is it? (Namely describe it with words, such as the rotation of angle π 7 around the point 2 i ; of course, this is not the answer.) 1

Homework 2, due on Friday September 9 Problem 1. Let (X, d) be a metric space. a. Show that d(p, Q) d(p, Q ) d(q, Q ) for every P, Q, Q X. b. Conclude that d(p, Q) d(p, Q ) d(q, Q ) for every P, Q, Q X. Problem 2. Let X be the plane R 2, and let d 1, d 2, d 3 : X X R be defined by d 1 ( (x, y), (x, y ) ) = (x x ) 2 + (y y ) 2 d 2 ( (x, y), (x, y ) ) = x x + y y d 3 ( (x, y), (x, y ) ) = max{ x x, y y }. In particular, d 1 is the usual euclidean distance d euc, and we proved in class that (X, d 1 ) is a metric space. a. Show that (X, d 2 ) is a metric space. b. Show that (X, d 3 ) 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 B d (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, d 1 ), (X, d 2 ) and (X, d 3 ) of Problem 2 and for the point P 0 = (0, 0) in X = R 2, draw the balls B d1 (P 0, 1), B d2 (P 0, 1) and B d3 (P 0, 1).

Homework 3, due on Wednesday September 14 Problem 1. In the hyperbolic plane H 2, consider the two points P = i and Q = 4 + i. For u > 0, let P u = ui, let Q u = 4 + ui, and let γ u be the curve going from P to Q that is made up of the vertical line segment [P, P u ], followed by the horizontal line segment [P u, Q u ], and finally followed by the vertical segment [Q u, Q]. a. Draw a picture of γ u. b. Compute the hyperbolic length l hyp (γ u ). c. For which value of u is l hyp (γ u ) minimum? (Hint: Remember calculus?) d. Use Part c to show that d hyp (P, Q) 2 ln 2 + 2. Problem 2. Let ϕ: H 2 H 2 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 H 2 and if γ 1 is the image of γ under ϕ, then l hyp (γ 1 ) = l hyp (γ). b. Use Part a to show that ϕ is an isometry from (H 2, d hyp ) to itself.

Homework 4, due on Wednesday September 21 Problem 1. Given four numbers a, b, c, d C with ad bc = 1 consider the map ϕ(z) = az + b cz + d defined for any complex number z different from d c. a. Given a similar map b. If ϕ (z) = a z + b c z + d with a, b, c, d C and a d b c = 1, compute the composition ϕ ϕ and show that there exists a, b, c, d C with a d b c = 1 such that ϕ ϕ (z) = a z + b c z + d for every z where it is defined. ψ(z) = compute ϕ ( ψ(z) ) and ψ ( ϕ(z) ). dz b cz + a, Remark. (No credit) If you remember from linear algebra how to multiply matrices, you may notice that (a ) ( ) ( ) b a b a b c d = c d c d. This is not a coincidence. (Do not write anything. This is just intended to whet your appetite for more math.) Problem 2. Let ϕ(z) = az + b with a, b, c, d R and ad bc = 1 cz + d as in Problem 1, and suppose in addition that a 0. Note that a, b, c, d are now real numbers. Set ϕ 1 (z) = z + b a ϕ 2 (z) = 1 z ϕ 3 (z) = 1 a 2 z ϕ 4(z) = z + c a. a. Which ones of ϕ 1, ϕ 2, ϕ 3, ϕ 4 are horizontal translations, homotheties or inversions? b. Show that ϕ = ϕ 2 ϕ 4 ϕ 3 ϕ 2 ϕ 1. c. Show that ϕ defines an isometry of the hyperbolic plane (H 2, d hyp ). (Hint: Part b.)

Homework 5, due on Wednesday September 29 Problem 1. Inspired by what we did in class for isometries of the hyperbolic plane (H 2, d hyp ), the goal of this problem is to describe all isometries of the euclidean plane (R 2, d euc ). More precisely, we will rigorously prove that all isometries of (R 2, d euc ) 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 P 1 = (0, 0) and P 2 = (1, 0). Show that, for any two positive numbers d 1 and d 2, there exists exactly zero, one or two points P = (x, y) such that d euc (P, P 1 ) = d 1 and d euc (P, P 2 ) = d 2. When they are two such points, show that they are related to each other by reflection across the x axis. (Hint: Express d euc (P, P 1 ) and d euc (P, P 2 ) in terms of x and y, and solve.) b. Consider in addition the point P 3 = (0, 1). Show that if the two points P = (x, y) and P = (x, y ) are such that d euc (P, P 1 ) = d euc (P, P 1 ), d euc (P, P 2 ) = d euc (P, P 2 ) and d euc (P, P 3 ) = d euc (P, P 3 ), then necessarily P = P. c. Let ϕ: R 2 R 2 be an isometry of (R 2, d euc ) such that ϕ(p 1 ) = P 1, ϕ(p 2 ) = P 2 and ϕ(p 3 ) = P 3. Show that ϕ(p ) = P for every P R 2. d. Let ϕ: R 2 R 2 be an isometry of (R 2, d euc ) such that ϕ(p 1 ) = P 1 and ϕ(p 2 ) = P 2. 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: R 2 R 2 be an isometry of (R 2, d euc ). Show that there exists a translation ψ 1 that sends ϕ(p 1 ) to P 1. Show that there exists a rotation ψ 2 around the point P 1 = (0, 0) such that ψ 2 (ψ 1 ( ϕ(p2 ) )) = P 2. (You may need to use the fact that ψ 1 and ϕ are isometries.) Show that, for the composition ψ = ψ 2 ψ 1, there exists z 1 C and an angle θ 1 R such that, in complex coordinates, ψ(z) = e iθ1 z + z 1. f. For ϕ and ψ as in Part e, show that the composition ψ ϕ is an isometry of (R 2, d euc ) that sends P 1 to P 1, and sends P 2 to P 2. g. Combine Parts d, e and f (and a short computation) to show that, for every isometry ϕ of the euclidean plane (R 2, d euc ), there exists z 0 C and θ R such that, either ϕ(z) = e iθ z + z 0 for every z C, or ϕ(z) = e 2iθ z + z 0 for every z C. Problem 2. Problem 1 is long enough. There is no Problem 2.

Math 434 Practice Midterm 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 (H 2, d hyp ). a. Compute the hyperbolic length l 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 l hyp )? c. Give a parametrization of this shortest curve from P to Q. d. Compute the hyperbolic distance d hyp (P, Q). Problem 2. On a set X, define for every two points P, Q X. d(p, Q) = { 0 if P = Q 1 if P Q 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(p 0 ) = P 0 for some point P 0 X. Show that ϕ sends each P X to a point ϕ(p ) that is at the same distance from P 0 as P, namely such that d ( ϕ(p ), P 0 ) = d(p, P0 ). Problem 4. a. Show that (cos θ + i sin θ) 5 = cos 5θ + i sin 5θ for every θ R. Hint: e iθ. b. Use Part a to show that for every θ R. cos 5θ = cos 5 θ 10 cos 3 θ sin 2 θ + 5 cos θ sin 4 θ Problem 5. Let ϕ: H 2 H 2 be the isometry of (H 2, d hyp ) defined by ϕ(z) = az + b with a, b, c, d R and ad bc = 1. cz + d Suppose in addition that a + d > 2 and c 0. 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 H 2 such that ϕ(g) = g.

Math 434 Actual midterm 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 complex number z = 2e i π 4. b. (10%) Find r and θ such that 1 + i 3 = re iθ. Problem 2. (Total: 30%) Consider the points P = (1, 3) and Q = (0, 2) in the hyperbolic plane (H 2, d hyp ). 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 l hyp )? c. (8%) Give a parametrization of this shortest curve from P to Q. d. (8%) Express the hyperbolic distance d hyp (P, Q) as an integral of explicit functions, but do not try to compute this integral. (Namely, leave your answer as something like d hyp (P, Q) = 13 π 7 ln t3 + cos 5t sin 3 t + 5 dt.) 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 z y} if x < y d(x, y) = sup{f(z); y z 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: H 2 H 2 be the (antilinear fractional) isometry of (H 2, d hyp ) defined by ψ(z) = c z + d with a, b, c, d R and ad bc = 1. a z + b Suppose in addition that b + c = 0 and a 0. Show that the set of points z H 2 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.)

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 B d (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 (B 2, d B 2). Let 0 be the center of the disk B 2. ) a. For a point P B 2, express the B 2 length l B 2( [0, P ] of the line segment [0, P ] in terms of the euclidean distance D = d euc (0, P ). In the, I know, very unlikely event that you forgot about partial fractions I remind you that dx 1 x 2 = 1 2 dx 1 + x + 1 2 dx 1 x = 1 2 ln 1 + x 1 x + C. 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 d B 2(0, P )? c. Show that the ball B db 2 (0, r) in B 2 coincides with the euclidean open disk of radius tanh r 2 = e r 2 e r 2 e r 2 +e r 2 centered at 0. Problem 2. We now consider the hyperbolic plane (H 2, d hyp ), and the isometry Φ: H 2 B 2 from (H 2, d hyp ) to (B 2, d B 2) defined by Φ(z) = z + i z + i. Also consider the linear fractional map Ψ defined by 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 B 2 to a point of H 2, and defines an isometry from (B 2, d B 2) to (H 2, d hyp ). b. Use Part a to show that, for the point i = Ψ(0), the ball B dhyp (i, r) is the image of the ball B db 2 (0, r) under Ψ, namely that B dhyp (i, r) = Ψ ( B db 2 (0, r) ). c. Use Problem 1 and a certain property of linear fractional maps to show that B dhyp (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 r 2 ) = e r, Ψ( tanh r 2 ) = er, and is orthogonal to the y axis. e. Show that the ball B dhyp (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, I remind you that cosh r = er +e r 2 and sinh r = er e r 2. f. Show that, for every y > 0, the ball B dhyp (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 H 2, the ball B dhyp (z, r) is an open euclidean disk. What are its euclidean center and euclidean radius? (Hint: horizontal translation.) Problem 3. In the sphere S 2, let N = (0, 0, 1) be the North Pole. Describe each of the balls B dsph (N, π 2 ), B d sph (N, π), B dsph (N, 3π 2 ) and B d sph (N, 2π) with a picture and a few words.

Homework 7, due on Wednesday October 26 Problem 1. Remember that the metric space (X, d) is locally isometric to the metric space (X, d ) if, for every P X, there exists an isometry ϕ: B d (P, r) B d (P, r) from a small ball B d (P, r) centered at P in X and a small ball B d (P, r) in X. Also, (X, d) is locally homogeneous if, for every P, Q X, there exists an isometry ϕ: B d (P, r) B d (Q, r) from a small ball B d (P, r) centered at P in X to a small ball B d (Q, r) centered at Q. Show that, if (X, d) is locally isometric to (X, d ) and if (X, d ) is locally homogeneous, then (X, d) is locally homogeneous. Problem 2. In the plane X = R 2, consider for each c R the hyperbola H c = {(x, y) R 2 ; xy = c}. (When c = 0, the hyperbola H 0 is somewhat degenerate.) a. Draw a picture of H 1, H 1, H 0 and H 1. 2 b. Show that the hyperbolas H c form a partition X of X = R 2, in the sense that every point P R 2 belongs to one and only one hyperbola H c. c. Consider the hyperbolas H c1 and H c2 associated to positive numbers c 1, c 2 > 0. Show that, for every ε > 0, there exist two points P 1 H c1 and P 2 H c2 such that d euc (P 1, P 2 ) < ε. d. More generally, consider the hyperbolas H c1 and H c2 associated to arbitrary numbers c 1, c 2 R. Show that, for every ε > 0, there exist two points P 1 H c1 and P 2 H c2 such that d euc (P 1, P 2 ) < ε. e. Let d euc be the quotient semi-metric on the partition X defined (using discrete walks as seen in class) by the euclidean metric d euc of X = R 2. In particular, for P R 2, let P X denote the hyperbola H c that contains it. (i) Show that d euc ( P 1, P 2 ) d euc (P 1, P 2 ), for every P 1 H c1 and P 2 H c2. (Hint: Can you find a discrete walk from P 1 to P 2?) (ii) Conclude that d euc ( P 1, P 2 ) = 0 for every P 1, P2 X. Hint: Part d. (iii) Is ( X, d euc ) a metric space?

Homework 8, due on Wednesday November 3 Recall from class that a homeomorphism from the metric space (X, d) to the metric space (X, d ) is a bijective map ϕ: X X 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 (R 2, d euc ), and let ( X, 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 X? b. Is the quotient space ( X, d euc ) locally isometric to the euclidean plane (R 2, d euc )? Explain. c. Give a proof by pictures, like the ones we have used in class in recent weeks, suggesting that the quotient space ( X, 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 R 2, 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) R 2 ; 0 x 1, 0 y 1} in the euclidean plane, and let ( X, 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 = 1 2 } in X. Draw a picture of α. Show that its image ᾱ in X is a closed curve, namely that its end points are glued together. b. For ᾱ as in Part a, let X ᾱ consists of all points P X that are not in ᾱ. What is X ᾱ homeomorphic to? (Use a proof by picture.) c. Let β be the vertical line segment {(x, y) X; x = 1 2 } in X. Draw a picture of β. Show that its image β is a closed curve in X. What is X β homeomorphic to? (Use a proof by picture.) d. Let γ consist of the two vertical line segments {(x, y) X; x = 1 3 or 2 3 } in X. Draw a picture of γ. Show that the image γ consists of a single closed curve in X. What is X β homeomorphic to? (Use a proof by picture.)

, Practice Final Exam Problem 1. (10%) Sketch a tessellation of the euclidean plane R 2 by triangles whose angles are π 2, π 3, π 6. Problem 2. Let X be a regular dodecagon in the hyperbolic plane (H 2, d hyp ), with all 12 sides of equal lengths and all 12 angles equal to θ. Label the vertices of X as V 1, V 2,..., V 12 in this order around X, and glue the edge A 1 A 2 to A 8 A 7, the edge A 2 A 3 to A 1 A 12, the edge A 3 A 4 to A 6 A 5, the edge A 4 A 5 to A 11 A 10, the edge A 6 A 7 to A 9 A 8, and the edge A 9 A 10 to A 12 A 11. (It may help to draw arrows on the picture below, which represents X is the disk model for symmetry.) Let ( X, d hyp ) be the corresponding quotient space. A 12 A 1 A 11 A 2 A 10 A 3 A 9 A 4 A 8 A 5 A 7 a. (8%) How many points of X correspond to the vertices of X? b. (8%) For which value of θ is the quotient space ( X, d hyp ) locally isometric to the hyperbolic plane (H 2, d hyp )? A 6 Problem 3. (12%) Let X be a polygon in the euclidean plane R 2, and let X be the quotient space obtained by gluing edges of X together. Given two points P, Q X in this quotient space, give the precise definition of a discrete walk w from P to Q, and of the length l 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 0 if x = y d(x, y) = max{ 1 q ; p, q integers, q > 0, x < p q < y} if x < y max{ 1 q ; p, q integers, q > 0, y < p q < 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, 1 2 ), d(0, 1 3 ) and d( 1 2, 1 3 ).

b. (14%) Show that d is indeed a metric, and that (R, d) is a metric space. Problem 5. Consider a hyperbolic isometry ϕ(z) = az + b with a, b, c, d R, ad bc = 1 and c 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 { a c }, where C is a circle in C that is tangent to the real line R at the point a c, and where C { a c } denotes the circle C from which the point a c 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: H 2 H 2 be a hyperbolic isometry sending the point to itself. Show that, either ϕ is a horizontal translation ϕ(z) = z + x 0 with x 0 R, or there exists a complete geodesic g such that ϕ(g) = g. (Possible hint: Write ϕ as ϕ(z) = az+b cz+d or ϕ(z) = c z+d a z+b and look for the end points of g.)