Lecture Notes for 120 - UCLA Department of Mathematics

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$Math 33B/1 S00 K. Solutions to Midterm Exam G1, Version B If you ...$

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6.5. LOCAL GAUSS-BONNET 156 where the coefficient a in front of N is irrelevant for computing the inner product with S and hence the geodesic curvature.
CHAPTER 7 Other Topics We cover geodesics, constant curvature surfaces, and isometries. 7.1. Geodesics Definition 7.1.1. Acurveq is called a geodesic on the surface if the tangential part of the acceleration vanishes ¨q I =0, or specifically d 2 u dt 2 + u ( ˙q, ˙q) = 0, d 2 v dt 2 + v ( ˙q, ˙q) = 0. This is equivalent to saying that ¨q is normal to the surface or that ¨q = ¨q II = NII ( ˙q, ˙q). Proposition 7.1.2. A geodesic has constant speed. Proof. Let q (t) be a geodesic. We compute the derivative of the square of the speed: d dt I(˙q, ˙q) = d ( ˙q · ˙q) =2¨q · ˙q = 2II ( ˙q, ˙q) N · ˙q =0 dt since N and ˙q are perpendicular. Thus q has constant speed. ⇤ Note that we used the second fundamental form to give a simple proof of this result. It is desirable and indeed possible to give a proof that only refers to the first fundamental form. The key lies in showing that we have a product rule for I(˙q, ˙q) that works just inside the surface. Since I ¨q I , ˙q = ¨q · ˙q this is fairly obvious. The goal would be to do the calculation using only the first fundamental form and that definitely possible but takes bit more effort. Next we address existence of geodesics. Theorem 7.1.3. Given a point p = q (u 0 ,v 0 ) and a tangent vector V = V u @q @u (u 0,v 0 )+V v @q @v (u 0,v 0 ) 2 T p M there is a unique geodesic q (t) =q (u (t) ,v(t)) defined on some small interval t 2 ( ", ") with the initial values q (0) = p, ˙q (0) = V. Proof. The existence and uniqueness part is a very general statement about solutions to differential equations. In this case we note that in the (u, v) parameters 157
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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
Page 106 and 107: 5.1. CURVES ON SURFACES 106 Exercis
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Page 118 and 119: 5.4. PRINCIPAL CURVATURES 118 The f
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Page 124 and 125: 5.5. RULED SURFACES 124 Definition
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Page 134 and 135: 6.1. CALCULATING CHRISTOFFEL SYMBOL
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Page 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
Page 142 and 143: 6.3. ACCELERATION 142 Alternately t
Page 144 and 145: 6.4. THE GAUSS AND CODAZZI EQUATION
Page 150 and 151: 6.5. LOCAL GAUSS-BONNET 150 (b) Use
Page 152 and 153: 6.5. LOCAL GAUSS-BONNET 152 We furt
Page 154 and 155: 6.5. LOCAL GAUSS-BONNET 154 clockwi
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Page 160 and 161: 7.2. UNPARAMETRIZED GEODESICS 160 T
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
Page 176 and 177: A.4. DIFFERENTIAL EQUATIONS 176 The
Page 178 and 179: A.4. DIFFERENTIAL EQUATIONS 178 Nex
Page 180 and 181: APPENDIX B Special Coordinate Repre
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
Page 190: B.7. ISOTHERMAL COORDINATES 190 K =
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Lecture Notes for 120 - UCLA Department of Mathematics

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