Lecture Notes for 120 - UCLA Department of Mathematics

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4.6. THE GAUSS FORMULAS 98 and then the tangential part w 1w 2 = ⇥ @q @u @q @v ⇤ [I] 1 ⇥ @q @u @q @v ⇤ t @ 2 q @w 1 @w 2 Definition 4.6.1. The Christoffel symbols of the first kind are defined as w 1w 2w = @2 q · @q @w 1 @w 2 @w Definition 4.6.2. The second fundamental form with respect to the normal N is defined as II (X, Y ) = ⇥ X u X ⇤ apple Y v u [II] Y v where = ⇥ X u X ⇤ apple apple v L uu L uv Y u L vu L vv Y v L w1w 2 = @ 2 q @w 1 @w 2 · N The superscript N refers to the choice of normal and is usually suppressed since there are only two choices for the normal ±N. This also tells us that NII is independent of the normal. To further simplify expressions we also need to do the appropriate multiplication with g w4w5 Definition 4.6.3. The Christoffel symbols of the second kind are defined as apple w w 1w 2 = g wu w 1w 2u + g wv w 1w 2v, apple apple g uu g uv g vu g vv w 1w 2u u w 1w 2 v w 1w 2 = = [I] 1 ⇥ @q @u @q @v This now gives us the tangential component as w 1w 2 = ⇥ @q @u @q @v ⇤ [I] 1 ⇥ @q @u ⇤ t @q @v w 1w 2v @ 2 q @w 1 @w 2 ⇤ t @ 2 q @w 1 @w 2 u @q = w 1w 2 @u + v @q w 1w 2 @v The second derivatives of q (u, v) can now be expressed as follows in terms of the Christoffel symbols of the second kind and the second fundamental form. These are often called the Gauss formulas: @ 2 q @u 2 = @ 2 q @u@v = @ 2 q @v 2 = u uu u uv u vv @q @u + @q @u + @q @u + v @q uu v uv v vv @v + L uuN @q @v + L uvN = @2 q @v@u @q @v + L vvN
4.6. THE GAUSS FORMULAS 99 or or @ 2 q @w 1 @w 2 = @ ⇥ @q @w @u @q @v u w 1w 2 @q @u + ⇤ = ⇥ @q @u v w 1w 2 @q @v + L w 1w 2 N 2 @q @v N ⇤ 4 3 u u wu wv v v 5 wu wv L wu L wv This means that we have introduced notation for the first two columns in [D w ] . The last column is related to the last row and will also be examined in the next chapter. As we shall see, and indeed already saw when considering polar coordinates in the plane, these formulas are important for defining accelerations of curves. They are however also important for giving a proper definition of the Hessian or second derivative matrix of a function on a surface. This will be explored in an exercise later. The task of calculating the second fundamental form is fairly straightforward, but will be postponed until the next chapter. Calculating the Christoffel symbols is more complicated and is delayed until we’ve gotten used to the second fundamental form. Exercises. (1) Show that @N @w is always tangent to the surface. (2) Show that @ 2 q · N = @q · @N @w 1 @w 2 @w 2 @w 1 This shows that the derivatives of the normal can be computed knowing the first and second fundamental forms. (3) Show that [II] vanishes if and only if the normal vector is constant. Show in turn that this happens if and only if the surface is part of a plane. (4) Show that when q (u, v) is a Cartesian parametrization, i.e., apple 1 0 [I] = 0 1 then the Christoffel symbols vanish. Hint: This is not obvious since we don’t know @2 q @w 1@w 2 .
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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
Page 48 and 49: 2.5. TWO SURPRISING RESULTS 48 unit
Page 50 and 51: 2.6. CONVEX CURVES 50 Exercises. Po
Page 52 and 53: 2.6. CONVEX CURVES 52 Proof. Any cu
Page 54 and 55: 2.6. CONVEX CURVES 54 (9) Give an e
Page 56 and 57: 3.1. THE FUNDAMENTAL EQUATIONS 56 T
Page 58 and 59: 3.1. THE FUNDAMENTAL EQUATIONS 58 B
Page 60 and 61: 3.1. THE FUNDAMENTAL EQUATIONS 60 (
Page 62 and 63: 3.2. CHARACTERIZATIONS OF SPACE CUR
Page 64 and 65: 3.2. CHARACTERIZATIONS OF SPACE CUR
Page 66 and 67: 3.3. CLOSED SPACE CURVES 66 (12) Pr
Page 68 and 69: 3.3. CLOSED SPACE CURVES 68 Thus th
Page 70 and 71: 3.3. CLOSED SPACE CURVES 70 (b) Sho
Page 72 and 73: CHAPTER 4 Basic Surface Theory 4.1.
Page 74 and 75: 4.1. SURFACES 74 Definition 4.1.6.
Page 76 and 77: 4.1. SURFACES 76 (6) Many classical
Page 78 and 79: 4.2. TANGENT SPACES AND MAPS 78 Thi
Page 80 and 81: 4.2. TANGENT SPACES AND MAPS 80 We
Page 82 and 83: 4.2. TANGENT SPACES AND MAPS 82 Def
Page 84 and 85: 4.3. THE ABSTRACT FRAMEWORK 84 Befo
Page 86 and 87: 4.3. THE ABSTRACT FRAMEWORK 86 in o
Page 88 and 89: 4.4. THE FIRST FUNDAMENTAL FORM 88
Page 94 and 95: 4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 96 and 97: 4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 100 and 101: CHAPTER 5 The Second Fundamental Fo
Page 102 and 103: 5.1. CURVES ON SURFACES 102 This sh
Page 104 and 105: 5.1. CURVES ON SURFACES 104 We also
Page 106 and 107: 5.1. CURVES ON SURFACES 106 Exercis
Page 108 and 109: 5.2. THE GAUSS AND WEINGARTEN MAPS
Page 110 and 111: 5.2. THE GAUSS AND WEINGARTEN MAPS
Page 112 and 113: 5.3. THE GAUSS AND MEAN CURVATURES
Page 118 and 119: 5.4. PRINCIPAL CURVATURES 118 The f
Page 120 and 121: 5.4. PRINCIPAL CURVATURES 120 By le
Page 122 and 123: 5.5. RULED SURFACES 122 (16) Let q
Page 124 and 125: 5.5. RULED SURFACES 124 Definition
Page 126 and 127: i.e., d dv and X are parallel. In p
Page 128 and 129: 5.5. RULED SURFACES 128 This shows
Page 130 and 131: 5.5. RULED SURFACES 130 (b) Show th
Page 132 and 133: 5.5. RULED SURFACES 132 (e) Show th
Page 134 and 135: 6.1. CALCULATING CHRISTOFFEL SYMBOL
Page 136 and 137: 6.1. CALCULATING CHRISTOFFEL SYMBOL
Page 138 and 139: 6.2. GENERALIZED AND ABSTRACT SURFA
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
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6.4. THE GAUSS AND CODAZZI EQUATION
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6.5. LOCAL GAUSS-BONNET 150 (b) Use
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6.5. LOCAL GAUSS-BONNET 152 We furt
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6.5. LOCAL GAUSS-BONNET 154 clockwi
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6.5. LOCAL GAUSS-BONNET 156 where t
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7.1. GEODESICS 158 we must solve a
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7.2. UNPARAMETRIZED GEODESICS 160 T
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7.5. ISOMETRIES 162 since the v-cur
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7.6. THE UPPER HALF PLANE 164 Proof
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7.6. THE UPPER HALF PLANE 166 and t
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7.6. THE UPPER HALF PLANE 168 so so
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8.2. RIEMANNIAN GEOMETRY 170 The ke
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APPENDIX A Vector Calculus A.1. Vec
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A.2. GEOMETRY 174 the characteristi
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A.4. DIFFERENTIAL EQUATIONS 176 The
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A.4. DIFFERENTIAL EQUATIONS 178 Nex
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APPENDIX B Special Coordinate Repre
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This leaves us with finding L s s.
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B.3. MONGE PATCHES 184 So we immedi
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B.4. SURFACES GIVEN BY AN EQUATION
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B.6. CHEBYSHEV NETS 188 (1) Show th
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B.7. ISOTHERMAL COORDINATES 190 K =
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