MATH 434 Fall 2016 Homework 1, due on Wednesday August 31
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1 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
2 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).
3 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 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.
4 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.)
5 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.
6 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.
7 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.)
8 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 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.)
9 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.
10 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?
11 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.)
12 , 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 ).
13 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.)
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