Blaga P. Lectures on the differential geometry of - tiera.ru

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44 Chapter 1. Space curves Remark. Be careful, the fact that a parameterized curve has zero curvature simply means that the support of the curve lies on a straight line, but it doesn’t necessarily means that the parameterized curve is (the restriction of) an affine map from R to R 3 , nor that it is equivalent to such a particular parameterized curve. We can consider, as we did before, the parameterized curve r : R → R 3 , r = r0 + at 3 , where a is a constant, nonvanishing, vector from R 3 . Then we have, immediately, r ′ (t) = 2at 2 and r ′′ (t) = 3at, which means that the velocity and the acceleration of the curve are parallel, hence the curve has zero curvature but, as we also saw earlier, this parameterized curve is not equivalent to an affine parameterized curve. 1.7.1 The geometrical meaning of curvature Let us consider a naturally parameterized curve (I, r = r(s)). We denote by ∆ϕ(s) the measure of the angle between the versors r(s) and r(s + ∆s). Then � � � �r(s + ∆s) − r(s)� = 2 � ∆ϕ(s) � � �sin � � 2 � . Therefore, k(s) = �r ′′ � � (s)� = � � � lim � � � r(s + ∆s) − r(s) � 2 � � � ∆s→0 ∆s � = lim ∆s→0 |∆s| � � � = lim �∆ϕ(s) � � � ∆s→0 � � ∆s � · � � � �sin ∆ϕ(s) � � � 2 � � � � � � � � � = lim �∆ϕ(s) � � � � ∆s→0 � � � ∆s � = � � � �dϕ � � � � � ds � . � ∆ϕ(s) 2 �sin ∆ϕ(s) 2 Thus, if we have in mind that ∆ϕ(s) is the measure of the angle between the tangents to the curve at s and s + ∆s, the last formula gives us: Proposition 1.7.2. The curvature of a curve is the speed of rotation of the tangent line to the curve, when the tangency point is moving along the curve with unit speed. 1.8 The Frenet frame (the moving frame) of a parameterized curve At each point of the support of a biregular parameterized curve (I, r = r(t)) one can construct a frame of the space R 3 . The idea is that, if we want to investigate the local properties of a parameterized curve around a given point of the curve, that it might be easier to do that if we don’t use the standard coordinate system of R 3 , but a coordinate � � � � =
1.8. The Frenet frame 45 system with the origin at the given point of the curve, while the coordinate axes have some connection with the local properties of the curve. Such a coordinate system was constructed, independently, at the middle of the XIX-th century, by the French mathematicians Frenet and Serret. Definition 1.8.1. The Frenet frame (or the moving frame) of a biregular parameterized curve (I, r = r(t)) at the point t0 ∈ I is an orthonormal frame of the space R 3 , with the origin at the point r(t0), the coordinate versors being the vectors {τ(t0), ν(t0), β(t0)}, where: • τ(t0) is the versor of the tangent to the curve at t0, i.e. τ(t0) = r′ (t0) �r ′ (t0)� . τ(t0) is also called the unit tangent at the point t0; • ν(t0) = k(t0)/k(t0) is the versor of the curvature vector: ν(t0) = k(t0) k(t0) and it is called the unit principal normal at the point t0. • β(t0) = τ(t0) × ν(t0) it is called the unit binormal at the point t0. • The axis of direction τ(t0) is, obviously, the tangent to the curve at t0. • The axis of direction ν(t0) is called the principal normal. In fact, this straight line is contained into the normal plane (since it is perpendicular on the tangent), but it is also contained into the osculating plane. Thus, the principal normal is the normal contained into the osculating plane. • The axis of direction β(t0) is called the binormal. The binormal is the normal perpendicular on the osculating plane. • The plane determined by the vectors {τ(t0), ν(t0} is the osculating plane (O in the figure 1.8). • The plane determined by the vectors {ν(t0), β(t0)} is the normal plane (N in the figure 1.8).
Page 1: Lectures on the Di
Page 5 and 6: Contents Foreword 9 I Curves 11 1 S
Page 7 and 8: Contents 7 4.5 The tangent vector s
Page 9 and 10: Foreword This book is based on the
Page 11: Part I Curves
Page 14 and 15: 14 Chapter 1. Space curves from the
Page 16 and 17: 16 Chapter 1. Space curves Definiti
Page 18 and 19: 18 Chapter 1. Space curves Figure 1
Page 20 and 21: 20 Chapter 1. Space curves (I, r) i
Page 22 and 23: 22 Chapter 1. Space curves Remark.
Page 24 and 25: 24 Chapter 1. Space curves and ρ
Page 26 and 27: 26 Chapter 1. Space curves given by
Page 28 and 29: 28 Chapter 1. Space curves • a sm
Page 30 and 31: 30 Chapter 1. Space curves Implicit
Page 34 and 35: 34 Chapter 1. Space curves Theorem
Page 36 and 37: 36 Chapter 1. Space curves Explicit
Page 38 and 39: 38 Chapter 1. Space curves From thi
Page 40 and 41: 40 Chapter 1. Space curves Theorem
Page 42 and 43: 42 Chapter 1. Space curves 1.7 The
Page 48 and 49: 48 Chapter 1. Space curves 1.9 Orie
Page 50 and 51: 50 Chapter 1. Space curves 1.10 The
Page 52 and 53: 52 Chapter 1. Space curves therefor
Page 54 and 55: 54 Chapter 1. Space curves Proposit
Page 56 and 57: 56 Chapter 1. Space curves or 1 + r
Page 58 and 59: 58 Chapter 1. Space curves or c0 ·
Page 60 and 61: 60 Chapter 1. Space curves Let ω b
Page 62 and 63: 62 Chapter 1. Space curves Corollar
Page 64 and 65: 64 Chapter 1. Space curves vector r
Page 66 and 67: 66 Chapter 1. Space curves a) If a
Page 68 and 69: 68 Chapter 1. Space curves d) We sh
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Page 72 and 73: 72 Chapter 1. Space curves 1.14.3 T
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Page 76 and 77: 76 Chapter 2. Plane curves Proof. I
Page 78 and 79: 78 Chapter 2. Plane curves By subst
Page 80 and 81: 80 Chapter 2. Plane curves The foll
Page 82 and 83: 82 Chapter 2. Plane curves c) J(Jv)
Page 84 and 85: 84 Chapter 2. Plane curves therefor
Page 86 and 87: 86 Chapter 2. Plane curves whence,
Page 88 and 89: 88 Chapter 2. Plane curves Figure 2
Page 90 and 91: 90 Chapter 2. Plane curves whence w
Page 92 and 93: 92 Chapter 2. Plane curves where A(
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94 Chapter 2. Plane curves
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96 Chapter 3. The integration of th
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98 Chapter 3. The integration of th
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100 Chapter 3. The integration of t
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104 Problems 6. A straight line seg
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106 Problems 19. Show that the Bern
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108 Problems 35. Find the envelopes
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110 Problems d) x 2 y 2 = (a 2 −
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4.1 Parameterized surfaces (patches
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4.2. Surfaces 117 Explicit represen
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4.3. The equivalence of local param
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4.3. The equivalence of local param
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4.5. The tangent vector space, the
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4.5. The tangent vector space, the
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4.6. The orientation of surfaces 12
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and 4.6. The orientation of surface
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4.7. Differentiable maps on a surfa
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4.7. Differentiable maps on a surfa
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where 4.8. The differential of a sm
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4.9. The spherical map and the shap
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4.10. The first fundamental form of
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4.11. The matrix of the shape opera
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4.12. The second fundamental form o
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4.13. The normal curvature. The Meu
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4.14. Asymptotic directions and asy
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4.15. The classification of points
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4.15. The classification of points
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4.16. Principal directions and curv
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4.17. The fundamental equations of
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4.18. The Gauss’ egregium theorem
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4.19 Geodesics 4.19.1 Introduction
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4.19. Geodesics 177 will denote it
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4.19. Geodesics 179 where, as we kn
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Examples of geodesics 4.19. Geodesi
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4.19. Geodesics 183 Here, for the f
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5.1 Ruled surfaces CHAPTER 5 Specia
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5.1. Ruled surfaces 187 Figure 5.2:
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5.1. Ruled surfaces 189 Proof. Sinc
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5.1. Ruled surfaces 191 surfaces, g
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5.1. Ruled surfaces 193 The three c
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5.1. Ruled surfaces 195 We can rega
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5.1. Ruled surfaces 197 5.1.5 Devel
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5.1. Ruled surfaces 199 • The str
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5.2. Minimal surfaces 201 Given a c
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5.2. Minimal surfaces 203 The follo
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5.2. Minimal surfaces 205 Proof. Le
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5.2.3 Ruled minimal surfaces 5.2. M
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5.2. Minimal surfaces 209 asymptoti
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5.2. Minimal surfaces 211 as γ is
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5.3. Surfaces of constant curvature
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5.3. Surfaces of constant curvature
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1. We consider, in the coordinate p
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Problems 219 12. Show that the norm
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Problems 221 33. Find the envelope,
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Problems 223 45. Find the equation
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Problems 225 58. Find the second fu
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Problems 227 67. Through a point M
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[1] Bär, C. - Elementare Different
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Bibliography 231 [29] Jellet, J.H.
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affine normal, 109 arc length, 19,
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182, 185, 187-189, 202, 205, 213, 2
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torus, 118, 119, 154, 191, 217, 219
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Blaga P. Lectures on the differential geometry of - tiera.ru

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