HYPERBOLIC GEOMETRY
CONTENTS 1 THE BASIC SPACES A MODEL FOR THE HYPERBOLIC PLANE THE RIEMANN SPHERE C THE BOUNDARY AT IN?NITY OF H 2 THE GENERAL M¨OBIUS GROUP THE GROUP OF M¨OBIUS TRANSFORMATIONS TRANSITIVITY PROPERTIES OF M¨OB+ THE CROSS RATIO CLASSI?CATION OF M¨OBIUS TRANSFORMATIONS A MATRIX REPRESENTATION RE?ECTIONS THE CONFORMALITY OF ELEMENTS OF M¨OB PRESERVING H TRANSITIVITY PROPERTIES OF M¨OB(H) 2 10 THE GEOMETRY OF THE ACTION OF M¨OB(H) 3 LENGTH AND DISTANCE IN H PATHS AND ELEMENTS OF ARC-LENGTH THE ELEMENT OF ARC-LENGTH ON H PATH METRIC SPACES FROM ARC-LENGTH TO METRIC FORMULAE FOR HYPERBOLIC DISTANCE IN H ISOMETRIES METRIC PROPERTIES OF (H, DH) 4 PLANAR MODELS OF THE HYPERBOLIC PLANE THE POINCAR´E DISC MODEL A GENERAL CONSTRUCTION 5 CONVEXITY, AREA, AND TRIGONOMETRY CONVEXITY HYPERBOLIC POLYGONS THE DE?NITION OF HYPERBOLIC AREA AREA AND THE GAUSS–BONNET FORMULA APPLICATIONS OF THE GAUSS–BONNET FORMULA TRIGONOMETRY IN THE HYPERBOLIC PLANE 6 NONPLANAR MODELS THE HYPERBOLOID MODEL OF THE HYPERBOLIC PLANE HIGHER DIMENSIONAL HYPERBOLIC SPACES SOLUTIONS TO EXERCISES REFERENCES THE BASIC SPACES IN THIS CHAPTER, WE SET THE STAGE FOR WHAT IS TO COME NAMELY, WE DE?NE THE UPPER HALF-PLANE MODEL H OF THE HYPERBOLIC PLANE, WHICH IS WHERE MUCH OF THE ACTION IN THIS BOOK TAKES PLACE WE DE?NE HYPERBOLIC LINES AND TALK A BIT ABOUT PARALLELISM TO AID OUR CONSTRUCTION OF A REASONABLE GROUP OF TRANSFORMATIONS OF H, WE EXPAND OUR HORIZONS TO CONSIDER THE RIEMANN SPHERE C AND CLOSE THE CHAPTER BY CONSIDERING HOW H SITS AS A SUBSET OF C 1 1 A MODEL FOR THE HYPERBOLIC PLANE WE BEGIN OUR INVESTIGATION BY DESCRIBING A MODEL OF THE HYPERBOLIC PLANE BY A MODEL, WE MEAN A CHOICE OF AN UNDERLYING SPACE, TOGETHER WITH A CHOICE OF HOW TO REPRESENT BASIC GEOMETRIC OBJECTS, SUCH AS POINTS AND LINES, IN THIS UNDERLYING SPACE AS WE SHALL SEE OVER THE COURSE OF THE BOOK, THERE ARE MANY POSSIBLE MODELS FOR THE HYPERBOLIC PLANE, EACH USEFUL IN ITS CONTEXT WE FOCUS OUR ATTENTION PRIMARILY, ALTHOUGH NOT EXCLUSIVELY, ON MODELS OF THE HYPERBOLIC PLANE WHOSE UNDERLYING SPACE IS CONTAINED IN THE COMPLEX PLANE C WE HAVE CHOSEN THE MODELS WE WORK WITH AND DESCRIBE IN THIS BOOK FOR THEIR CONVENIENCE TO GIVE