Lecture Notes for Differential Geometry
                           James S. Cook
                          Liberty University
                       Department of Mathematics
                            Summer 2015
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                  Contents
                  1 introduction                                                                                               5
                     1.1   points and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        5
                     1.2   on tangent and cotangent spaces and bundles . . . . . . . . . . . . . . . . . . . . . .             7
                     1.3   the wedge product and differential forms . . . . . . . . . . . . . . . . . . . . . . . . .          11
                           1.3.1    the exterior algebra in three dimensions . . . . . . . . . . . . . . . . . . . . .        15
                     1.4   paths and curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       16
                     1.5   the push-forward or differential of a map . . . . . . . . . . . . . . . . . . . . . . . . .         19
                  2 curves and frames                                                                                        25
                     2.1   on distance in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        26
                     2.2   vectors and frames in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . .         27
                     2.3   calculus of vectors fields along curves . . . . . . . . . . . . . . . . . . . . . . . . . . .       33
                     2.4   Frenet Serret frame of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       35
                           2.4.1    the non unit-speed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       41
                     2.5   covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      44
                     2.6   frames and connection forms        . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   48
                           2.6.1    on matrices of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . .      49
                     2.7   coframes and the Structure Equations of Cartan            . . . . . . . . . . . . . . . . . . . .  53
                  3 euclidean geometry                                                                                       57
                     3.1   isometries of euclidean space      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   58
                     3.2   how isometries act on vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        61
                           3.2.1    Frenet curves in Rn     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   63
                     3.3   on frames and congruence in three dimensions . . . . . . . . . . . . . . . . . . . . . .           67
                  introduction and motivations for these notes
                  Certainly many excellent texts on differential geometry are available these days. These notes most
                  closely echo Barrett O’neill’s classic Elementary Differential Geometry revised second edition. I
                  taught this course once before from O’neil’s text and we found it was very easy to follow, however,
                  I will diverge from his presentation in several notable ways this summer.
                     1. I intend to use modern notation for vector fields. I will resist the use of bold characters as I
                        find it frustrating when doing board work or hand-written homework.
                     2. I will make some general n-dimensional comments here and there. So, there will be two tracks
                        in these notes: first, the direct extension of typical American third semester calculus in R3
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                        (with the scary manifold-theoretic notation) and second, some results and thoughts for the
                        n-dimensional context.
                 I hope to borrow some of the wisdom of Wolfgang Kuhnel’s¨        Differential Geometry: Curves-Surfaces-
                 Manifolds. I think the purely three dimensional results are readily acessible to anyone who has
                 taken third semester calculus. On the other hand, general n-dimensional results probably make
                 more sense if you’ve had a good exposure to abstract linear algebra.
                 I do not expect the student has seen advanced calculus before studying these notes. However, on
                 the other hand, I will defer proofs of certain claims to our course in advanced calculus.