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¨‖2TN 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
欢迎咨询51作业君