Notes on Non-Euclidean Geometry
                    Jiayin Pan
                     Abstract. Lecture notes for Math 113 Non-Euclidean geometry. We loosely
                     follow the textbook Geometries and Groups by Nikulin and Shafarevich. Most
                     proofs have been rewritten and more content has been added. This note is self-
                     contained.
                       Last updated: December 14, 2020
                                                            Contents
                            Chapter 1.  Motivating examples                                             5
                              1.1.  Sphere                                                              5
                              1.2.  Cylinder                                                            7
                              1.3.  More examples                                                      10
                            Chapter 2.  Basic Concepts                                                 13
                              2.1.  Geometries                                                         13
                              2.2.  Groups                                                             16
                              2.3.  Group actions on Geometries                                        18
                                                                              2    2
                            Chapter 3.  Free and discrete isometric actions on R or S                  23
                              3.1.  Hello again, linear algebra                                        23
                              3.2.  Elements of Isom(R2) without fixed points                           25
                              3.3.  Subgroups of Isom(R2)                                              26
                                                       2
                              3.4.  Subgroups of Isom(S )                                              30
                            Chapter 4.  Coverings                                                      33
                              4.1.  Covering maps                                                      33
                              4.2.  Cover a 2-dimensional locally Euclidean space by R2                35
                              4.3.  Covering groups                                                    37
                            Chapter 5.  Hyperbolic Plane                                               41
                              5.1.  Introduction                                                       41
                              5.2.  Pure imaginary half line under the SL (R)-action                   42
                                                                        2
                              5.3.  Distance function on H                                             44
                              5.4.  Projection                                                         48
                              5.5.  The isometry group of (H,d)                                        51
                                                                  3