Lecture Notes for 120 - UCLA Department of Mathematics

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7.2. UNPARAMETRIZED GEODESICS 160 Thus we have two equations ✓ du g uu dt + g vu Since dv dt ◆✓ d 2 u dt 2 + apple det ✓ dv d 2 u dt dt 2 + ◆ u ( ˙q, ˙q) g uu du dv dt = dv ✓ du g vu dt dt + g vv = | ˙q| 2 =1 the only possible solution is d 2 u dt 2 + showing that q is a geodesic. + dt + g uv dv dt dv dt ◆ u ( ˙q, ˙q) ✓ g uv du dt + g vv g vu du ◆ + du dt du dt ✓ du d 2 v dt dt 2 + ◆✓ d 2 v dt 2 + dv dt dt + g vv dv dt ✓ du g uu dt + g uv u ( ˙q, ˙q) =0= d2 v dt 2 + v ( ˙q, ˙q) , ◆ dv dt ◆ v ( ˙q, ˙q) ◆ v ( ˙q, ˙q) = 0, = 0 ⇤ Depending on our parametrization (u, v) geodesics can be pictured in many ways. We’ll study a few cases where geodesics take on some familiar shapes. Consider the sphere where great circles are described by ax + by + cz = 0, x 2 + y 2 + z 2 = 1 If we use the parametrization 1 p 1+s2 +t 2 (s, t, 1) , or in other words x z = s, y z = t then these equations simply become straight lines in (s, t) coordinates: as + bt + c =0 If we use the Monge patch u, v, p 1 u 2 v 2 then the equations become a 2 + c 2 u 2 +2abuv + b 2 + c 2 v 2 = c 2 These are the equations of ellipses whose axes go through the origin and are inscribed in the unit circle. This is how you draw great circles on the sphere! The first fundamental form is given by " u 1+ 2 [I] = # uv 1 u 2 v 2 1 u 2 v 2 1 u 2 v 2 uv v 1 u 2 v 1+ 2 2 Now consider an intrinsic metric on the (u, v) plane where we have simply made some sign changes from the unit sphere metric " [I] = # u 1 2 uv 1+u 2 +v 2 1+u 2 +v 2 uv v 1+u 2 +v 1 2 2 1+u 2 +v 2 Using the parameter independent approach to geodesics one can show that they turn out to be hyperbolas whose axes go through the origin a 2 c 2 u 2 +2abuv + b 2 c 2 v 2 = c 2
7.4. CONSTANT GAUSS CURVATURE 161 This metric can also be reparametrized to have its geodesics be straight lines. The later reparametrization is: u s = p 1+u2 + v 2 t = v p 1+u2 + v 2 and the geodesics given by as + bt + c =0. We shall later explicitly find the geodesics on the upper half plane using the equations developed here. Exercises. (1) Show that geodesics satisfy a second order equation of the type d 2 ✓ ◆ 3 ✓ ◆ 2 v dv dv du 2 = A + B + C dv du du du + D and calculate the functions A, B, C, D in terms of the appropriate Christoffel symbols and metric coefficients. 7.3. Shortest Curves The goal is to show that the shortest curves are geodesics, and conversely that sufficiently short geodesics are minimal in length. For the latter using geodesic coordinates makes an argument that is similar to the Euclidean version using a unit gradient field. There should be an easy extrinsic proof that the shortest curves are geodesics 7.4. Constant Gauss Curvature The goal will be to give a canonical local structure for surfaces with constant Gauss curvature. Given the plethora of surfaces with constant Gauss curvature we seek for the moment only canonical coordinates. Minding was the first person to give the classification of the first fundamental form we obtain below. Riemann extended this result to higher dimensions. We start by studying the case of vanishing Gauss curvature. Theorem 7.4.1. If a surface has zero Gauss curvature, then it admits Cartesian coordinates. Proof. We shall assume that we have geodesic coordinates along a unit speed geodesic. Thus the v-curve described by u =0is a geodesic, and by definition of geodesic coordinates all of the u-curves are unit speed geodesics. The first fundamental form is apple 1 0 [I] = 0 r 2 Assuming K =0, we immediately obtain r (u, v) =r (0,v)+u @r @u (0,v) But r (0,v)= @q @v =1
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Classical Differential Geometry Pet
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CONTENTS 4 Chapter 7. Other Topics
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1.1. CURVES 6 measurements are need
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1.1. CURVES 8 as we then get q (✓
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1.1. CURVES 10 these derivatives ar
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1.1. CURVES 12 (a) Show that if r (
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1.2. ARCLENGTH AND LINEAR MOTION 14
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1.3. CURVATURE 20 relates the side
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1.3. CURVATURE 22 of a that is norm
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1.3. CURVATURE 24 As the piece of t
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1.4. INTEGRAL CURVES 26 (14) Let q
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1.4. INTEGRAL CURVES 28 So at q 0 =
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CHAPTER 2 Planar Curves 2.1. Genera
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2.2. THE FUNDAMENTAL EQUATIONS 32 M
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2.2. THE FUNDAMENTAL EQUATIONS 34 (
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2.3. LENGTH AND AREA 36 (22) (Newto
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As ( 2.3. LENGTH AND AREA 38 sin
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2.3. LENGTH AND AREA 40 The choice
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2.4. THE ROTATION INDEX 42 the hypo
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2.4. THE ROTATION INDEX 44 where an
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2.4. THE ROTATION INDEX 46 Exercise
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2.5. TWO SURPRISING RESULTS 48 unit
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2.6. CONVEX CURVES 50 Exercises. Po
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2.6. CONVEX CURVES 52 Proof. Any cu
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2.6. CONVEX CURVES 54 (9) Give an e
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3.1. THE FUNDAMENTAL EQUATIONS 56 T
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3.1. THE FUNDAMENTAL EQUATIONS 58 B
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3.1. THE FUNDAMENTAL EQUATIONS 60 (
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.3. CLOSED SPACE CURVES 66 (12) Pr
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3.3. CLOSED SPACE CURVES 68 Thus th
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3.3. CLOSED SPACE CURVES 70 (b) Sho
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CHAPTER 4 Basic Surface Theory 4.1.
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4.1. SURFACES 74 Definition 4.1.6.
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4.1. SURFACES 76 (6) Many classical
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4.2. TANGENT SPACES AND MAPS 78 Thi
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4.2. TANGENT SPACES AND MAPS 80 We
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4.2. TANGENT SPACES AND MAPS 82 Def
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4.3. THE ABSTRACT FRAMEWORK 84 Befo
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4.3. THE ABSTRACT FRAMEWORK 86 in o
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4.4. THE FIRST FUNDAMENTAL FORM 88
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.6. THE GAUSS FORMULAS 98 and then
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CHAPTER 5 The Second Fundamental Fo
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5.1. CURVES ON SURFACES 102 This sh
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5.1. CURVES ON SURFACES 104 We also
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5.1. CURVES ON SURFACES 106 Exercis
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5.2. THE GAUSS AND WEINGARTEN MAPS
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Page 124 and 125: 5.5. RULED SURFACES 124 Definition
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Page 130 and 131: 5.5. RULED SURFACES 130 (b) Show th
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Page 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
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Page 152 and 153: 6.5. LOCAL GAUSS-BONNET 152 We furt
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Page 156 and 157: 6.5. LOCAL GAUSS-BONNET 156 where t
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Page 162 and 163: 7.5. ISOMETRIES 162 since the v-cur
Page 164 and 165: 7.6. THE UPPER HALF PLANE 164 Proof
Page 166 and 167: 7.6. THE UPPER HALF PLANE 166 and t
Page 168 and 169: 7.6. THE UPPER HALF PLANE 168 so so
Page 170 and 171: 8.2. RIEMANNIAN GEOMETRY 170 The ke
Page 172 and 173: APPENDIX A Vector Calculus A.1. Vec
Page 174 and 175: A.2. GEOMETRY 174 the characteristi
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Page 182 and 183: This leaves us with finding L s s.
Page 184 and 185: B.3. MONGE PATCHES 184 So we immedi
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Page 188 and 189: B.6. CHEBYSHEV NETS 188 (1) Show th
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Lecture Notes for 120 - UCLA Department of Mathematics

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