Geometry of Curves and Surfaces
                     Weiyi Zhang
              Mathematics Institute, University of Warwick
                   September 18, 2016
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                       Contents
                       1 Curves                                                                           5
                           1.1   Course description . . . . . . . . . . . . . . . . . . . . . . . .       5
                                 1.1.1   Abit preparation: Differentiation . . . . . . . . . . . .         6
                           1.2   Methods of describing a curve . . . . . . . . . . . . . . . . . .        8
                                 1.2.1   Fixed coordinates . . . . . . . . . . . . . . . . . . . . .      8
                                 1.2.2   Moving frames: parametrized curves . . . . . . . . . .           8
                                 1.2.3   Intrinsic way(coordinate free) . . . . . . . . . . . . . .       9
                           1.3   Curves in Rn: Arclength Parametrization . . . . . . . . . . .           10
                           1.4   Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     12
                           1.5   Orthonormal frame: Frenet-Serret equations . . . . . . . . . .          15
                           1.6   Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . .    17
                           1.7   More results for space curves . . . . . . . . . . . . . . . . . .       20
                                 1.7.1   Taylor expansion of a curve . . . . . . . . . . . . . . .       20
                                 1.7.2   Fundamental Theorem of the local theory of curves . .           21
                           1.8   Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . .      21
                           1.9   The Four Vertex Theorem . . . . . . . . . . . . . . . . . . . .         24
                       2 Surfaces in R3                                                                  29
                           2.1   Definitions and Examples . . . . . . . . . . . . . . . . . . . .         29
                                 2.1.1   Compact surfaces . . . . . . . . . . . . . . . . . . . . .      32
                                 2.1.2   Level sets . . . . . . . . . . . . . . . . . . . . . . . . .    33
                           2.2   The First Fundamental Form . . . . . . . . . . . . . . . . . .          34
                           2.3   Length, Angle, Area . . . . . . . . . . . . . . . . . . . . . . .       35
                                 2.3.1   Length: Isometry . . . . . . . . . . . . . . . . . . . . .      36
                                 2.3.2   Angle: conformal . . . . . . . . . . . . . . . . . . . . .      37
                                 2.3.3   Area: equiareal . . . . . . . . . . . . . . . . . . . . . .     37
                           2.4   The Second Fundamental Form . . . . . . . . . . . . . . . . .           38
                                 2.4.1   Normals and orientability . . . . . . . . . . . . . . . .       39
                                 2.4.2   Gauss map and second fundamental form . . . . . . .             40
                           2.5   Curvatures     . . . . . . . . . . . . . . . . . . . . . . . . . . . .  42
                                 2.5.1   Definitions and first properties . . . . . . . . . . . . .        42
                                 2.5.2   Calculation of Gaussian and mean curvatures . . . . .           45
                                 2.5.3   Principal curvatures . . . . . . . . . . . . . . . . . . .      47
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                                      2.6   Gauss’s Theorema Egregium           . . . . . . . . . . . . . . . . . .  49
                                            2.6.1    Gaussian curvature for special cases . . . . . . . . . .        52
                                      2.7   Surfaces of constant Gaussian curvature . . . . . . . . . . . .          53
                                      2.8   Parallel transport and covariant derivative . . . . . . . . . . .        56
                                      2.9   Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      58
                                            2.9.1    General facts for geodesics . . . . . . . . . . . . . . . .     58
                                            2.9.2    Geodesics on surfaces of revolution . . . . . . . . . . .       62
                                            2.9.3    Geodesics and shortest paths . . . . . . . . . . . . . .        64
                                            2.9.4    Geodesic coordinates . . . . . . . . . . . . . . . . . . .      65
                                            2.9.5    Half plane model of hyperbolic plane . . . . . . . . . .        67
                                      2.10 Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . .            68
                                            2.10.1 Geodesic polygons . . . . . . . . . . . . . . . . . . . .         70
                                            2.10.2 Global Gauss-Bonnet          . . . . . . . . . . . . . . . . . .  71
                                      2.11 Vector fields and Euler number . . . . . . . . . . . . . . . . .           73