Elementary Differential Geometry:
                   Lecture Notes
                  Gilbert Weinstein
                                                            Contents
                            Preface                                                                     5
                            Chapter 1.  Curves                                                          7
                              1.  Preliminaries                                                         7
                              2.  Local Theory for Curves in R3                                         8
                              3.  Plane Curves                                                         10
                              4.  Fenchel’s Theorem                                                    15
                              Exercises                                                                16
                            Chapter 2.  Local Surface Theory                                           19
                              1.  Surfaces                                                             19
                              2.  The First Fundamental Form                                           21
                              3.  The Second Fundamental Form                                          23
                              4.  Examples                                                             25
                              5.  Lines of Curvature                                                   28
                              6.  More Examples                                                        30
                              7.  Surface Area                                                         33
                              8.  Bernstein’s Theorem                                                  37
                              9.  Theorema Egregium                                                    39
                              Exercises                                                                41
                            Chapter 3.  Local Intrinsic Geometry of Surfaces                           45
                              1.  Riemannian Surfaces                                                  45
                              2.  Lie Derivative                                                       47
                              3.  Covariant Differentiation                                             48
                              4.  Geodesics                                                            50
                              5.  The Riemann Curvature Tensor                                         53
                              6.  The Second Variation of Arclength                                    56
                              Exercises                                                                59
                            Chapter 4.  Global Theory of Surfaces                                      61
                              1.  Differentiable Manifolds                                              61
                              2.  Some Multi-linear Algebra                                            64
                              3.  Integration                                                          65
                              Exercises                                                                65
                            Index                                                                      67
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