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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7.6. THE UPPER HALF PLANE 168 so so so h 2 +1= r2 v 2 ✓ ◆ 2 dv +1= r2 du v 2 r dv r du = ± 2 v 2 1 p r 2 v = ± 2 v 2 Even though this is a separable equation and the integrals involved can be computed it is still a mess to sort out. The answer however is fairly simple: q v = r 2 (u u 0 ) 2 . In other words the geodesics are either vertical lines or semicircles whose center is on the u axis. As these are precisely the curves that are fixed by mirror symmetries in vertical lines or inversions this should not be a big surprise. 7.6.3. Curvature of H. Having just computed the Christoffel symbols u uu = 1 @g 2 guu uu @u =0 v uu = 1 @g 2 gvv uu @v = 1 v v vv = 1 @g 2 gvv vv u vv = @v = 1 v 1 @g 2 guu vv @u =0 u uv = 1 @g 2 guu uu @v = 1 v v uv = 1 @g 2 gvv vv @u =0 it is now also possible to calculate the Riemannian curvature tensor @ u vu Rvvu u = @ u vv @u @v + u vv u uu + v vv u uv ( u vu u vu + v vu u vv) 1 ✓ ◆✓ ◆ ✓ ◆ @ 2 v 1 1 1 = 0 +0+ +0! @v v v v 1 = v 2 and the Gauss curvature K = Ru vvu g vv = 1 7.6.4. Conformal Picture. Triangles and angle sum. Parallel lines.
CHAPTER 8 Global Theory 8.1. Global Stuff Closed surfaces must have positive curvature somewhere. Convex surfaces. Constant mean curvature and/or Gauss curvature. Gauss-Bonnet. Hilbert. Theorem 8.1.1. A parametrized surface all of whose principal curvatures are ">0 is convex on regions of a fixed size depending on " and the domain. Proof. The important observation is that critical points for the height function are isolated, and, unlike the curve situation, the complement is connected! Also critical points are max or min by second derivative test. Probably need a domain B (0,R) ⇢ R 2 where the metric in polar coordinates has the form apple 1 0 [I] = 0 ⇢ 2 (r, ✓) and then restrict to B (0,"R) . ⇤ 8.2. Riemannian Geometry As with abstract surfaces we simply define what the dot products of the tangent fields should be: [I] = ⇥ @q @u 1 ··· ⇤ @q t ⇥ @q @u n @u ··· 1 2 ⇤ @q 6 @u = 4 n 3 g 11 ··· g 1n . . .. 7 . 5 g n1 ··· g nn The notation @q @u for the tangent field that corresponds to the velocity of the u i i curves is borrowed from our view of what happens on a surface. We have the very general formula for how vectors are expanded V = ⇥ ⇤ ⇣ ⇥ ⇤ t ⇥ ⇤ ⌘ 1 ⇥ ⇤ t E 1 ··· E n E1 ··· E n E1 ··· E n E1 ··· E n V 2 3 1 2 3 = ⇥ E 1 · E 1 ··· E 1 · E n E 1 · V ⇤ 6 E 1 ··· E n 4 . . .. 7 6 7 . 5 4 . 5 E n · E 1 ··· E n · E n E n · V provided we know how to compute dot products of the basis vectors and dots products of V with the basis vectors. 169
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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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5.2. THE GAUSS AND WEINGARTEN MAPS
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5.3. THE GAUSS AND MEAN CURVATURES
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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 132 and 133: 5.5. RULED SURFACES 132 (e) 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 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 170 and 171: 8.2. RIEMANNIAN GEOMETRY 170 The ke
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Page 174 and 175: A.2. GEOMETRY 174 the characteristi
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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
Page 186 and 187: B.4. SURFACES GIVEN BY AN EQUATION
Page 188 and 189: B.6. CHEBYSHEV NETS 188 (1) Show th
Page 190: B.7. ISOTHERMAL COORDINATES 190 K =
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