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

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154 Chapter 4. General theory of surfaces Elliptic points. At an elliptic point, the normal curvature has the same signs in all directions 4 . Applying the Meusnier’s theorem, this means that the centers of curvature of all normal sections lie on the same side of the surface. An example of surface that has only elliptic points is the ellipsoid, given, for instance, by the parameterization r(u, v) = (a cos u cos v, b sin u cos v, c sin v). (4.15.1) At an elliptic point there is no real asymptotic direction, therefore no asymptotic line passes through an elliptic point. Figure 4.7: Elliptic points on a surface Parabolic points. In this case the normal curvature does not change the sign, but there is exactly one direction where it vanishes. This is, clearly, an asymptotic direction. Thus, through a parabolic point of a surface passes only one asymptotic line. The cylinders and cones (with the apex removed) have only parabolic points. Hyperbolic points. In the case of hyperbolic points, it is possible for kn to change sign and there are exactly two direction where it vanishes. Thus, through a hyperbolic point of a surface pass two asymptotic lines. The hyperbolic points are also called saddle points. Such points we can find, for instance, on a hyperbolic paraboloid. There are, of course, many surfaces on which we can meet all three kind of points (for instance on the torus). 4 It is not necessarily positive.
4.15. The classification of points on a surface 155 Figure 4.8: The monkey saddle Figure 4.9: Hyperbolic points on a surface Flat points. The shape of the surface around a flat point might be quite complicated and it is difficult to study it. In fact, in many cases, when proving a theorem in surface theory one explicitly assumes that the surface has no flat points. The surface from the figure 4.15 (the monkey saddle has a flat point at the origin of coordinates.
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Lectures on the Di
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Contents Foreword 9 I Curves 11 1 S
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Contents 7 4.5 The tangent vector s
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Foreword This book is based on the
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Part I Curves
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14 Chapter 1. Space curves from the
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16 Chapter 1. Space curves Definiti
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18 Chapter 1. Space curves Figure 1
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20 Chapter 1. Space curves (I, r) i
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22 Chapter 1. Space curves Remark.
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24 Chapter 1. Space curves and ρ
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26 Chapter 1. Space curves given by
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28 Chapter 1. Space curves • a sm
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30 Chapter 1. Space curves Implicit
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34 Chapter 1. Space curves Theorem
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36 Chapter 1. Space curves Explicit
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38 Chapter 1. Space curves From thi
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40 Chapter 1. Space curves Theorem
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42 Chapter 1. Space curves 1.7 The
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44 Chapter 1. Space curves Remark.
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48 Chapter 1. Space curves 1.9 Orie
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50 Chapter 1. Space curves 1.10 The
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52 Chapter 1. Space curves therefor
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54 Chapter 1. Space curves Proposit
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56 Chapter 1. Space curves or 1 + r
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58 Chapter 1. Space curves or c0 ·
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60 Chapter 1. Space curves Let ω b
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62 Chapter 1. Space curves Corollar
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64 Chapter 1. Space curves vector r
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66 Chapter 1. Space curves a) If a
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68 Chapter 1. Space curves d) We sh
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70 Chapter 1. Space curves Then: τ
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72 Chapter 1. Space curves 1.14.3 T
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74 Chapter 1. Space curves
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76 Chapter 2. Plane curves Proof. I
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78 Chapter 2. Plane curves By subst
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80 Chapter 2. Plane curves The foll
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82 Chapter 2. Plane curves c) J(Jv)
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84 Chapter 2. Plane curves therefor
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86 Chapter 2. Plane curves whence,
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88 Chapter 2. Plane curves Figure 2
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90 Chapter 2. Plane curves whence w
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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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102 Chapter 3. The integration of t
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Page 106 and 107: 106 Problems 19. Show that the Bern
Page 108 and 109: 108 Problems 35. Find the envelopes
Page 110 and 111: 110 Problems d) x 2 y 2 = (a 2 −
Page 112 and 113: 112 Chapter 3. The integration of t
Page 115 and 116: 4.1 Parameterized surfaces (patches
Page 117 and 118: 4.2. Surfaces 117 Explicit represen
Page 119 and 120: 4.3. The equivalence of local param
Page 121 and 122: 4.3. The equivalence of local param
Page 123 and 124: 4.5. The tangent vector space, the
Page 125 and 126: 4.5. The tangent vector space, the
Page 127 and 128: 4.6. The orientation of surfaces 12
Page 129 and 130: and 4.6. The orientation of surface
Page 131 and 132: 4.7. Differentiable maps on a surfa
Page 133 and 134: 4.7. Differentiable maps on a surfa
Page 135 and 136: where 4.8. The differential of a sm
Page 137 and 138: 4.9. The spherical map and the shap
Page 139 and 140: 4.10. The first fundamental form of
Page 145 and 146: 4.11. The matrix of the shape opera
Page 147 and 148: 4.12. The second fundamental form o
Page 149 and 150: 4.13. The normal curvature. The Meu
Page 151 and 152: 4.14. Asymptotic directions and asy
Page 153: 4.15. The classification of points
Page 157 and 158: 4.16. Principal directions and curv
Page 163 and 164: 4.17. The fundamental equations of
Page 173 and 174: 4.18. The Gauss’ egregium theorem
Page 175 and 176: 4.19 Geodesics 4.19.1 Introduction
Page 177 and 178: 4.19. Geodesics 177 will denote it
Page 179 and 180: 4.19. Geodesics 179 where, as we kn
Page 181 and 182: Examples of geodesics 4.19. Geodesi
Page 183 and 184: 4.19. Geodesics 183 Here, for the f
Page 185 and 186: 5.1 Ruled surfaces CHAPTER 5 Specia
Page 187 and 188: 5.1. Ruled surfaces 187 Figure 5.2:
Page 189 and 190: 5.1. Ruled surfaces 189 Proof. Sinc
Page 191 and 192: 5.1. Ruled surfaces 191 surfaces, g
Page 193 and 194: 5.1. Ruled surfaces 193 The three c
Page 195 and 196: 5.1. Ruled surfaces 195 We can rega
Page 197 and 198: 5.1. Ruled surfaces 197 5.1.5 Devel
Page 199 and 200: 5.1. Ruled surfaces 199 • The str
Page 201 and 202: 5.2. Minimal surfaces 201 Given a c
Page 203 and 204: 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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