THE GEOMETRY OF METRIC SPACES Oona Rainio MSc Thesis November 2019 DEPARTMENT OF MATHEMATICS AND STATISTICS The originality of this thesis has been checked in accordance with the University of Turku quality assurance system using the Turnitin OriginalityCheck service UNIVERSITY OF TURKU Department of Mathematics and Statistics RAINIO, OONA: The Geometry of Metric Spaces MSc Thesis, 176 pages, 14 appendix pages, 51 figures Mathematics November 2019 This Master’s Thesis investigates the geometry of different metric spaces. The aim is to discuss the topic by giving several different metric spaces and illustrating their properties. While it is not possible to give all the information available on metric spaces and their geometry in this one thesis, we will introduce necessary definitions and several theorems needed to study this subject further. There are three types of metrics and metric spaces in our main focus. First, we will focus on the complex plane with the usual Euclidean metric that has been slightly modified so as to make it suitable for complex numbers. Then, we will present two different models for the hyperbolic geometry with their own hyperbolic distances that can be derived from each other. In the end of this thesis, we will also introduce the triangular ratio metric that can be applied to different domains in metric spaces that have fulfilled a certain condition presented with the definition. We will inspect different geometrical structures with the metrics mentioned above. Especially, we will concentrate on lines, line segments and triangles. Our focus is not only to inspect certain properties of these objects but also to find out how they are preserved in a specific group of transformations. Keywords: Complex number, complex plane, cross-ratio, geometry, hyperbolic ge- ometry, metric space, Möbius transformation, Poincaré disk model, triangular ratio metric, upper half-plane model. Contents 1 Introduction 1 2 Metric Spaces 3 2.1 Notion of a Metric . 3 2.2 Properties of a Metric Space . 13 2.3 Functions and Their Continuity . 18 3 Complex Plane 28 3.1 Introduction to Complex Numbers . 28 3.2 Complex Points and Lines . 36 3.3 Triangles in the Complex Plane . 48 3.4 Transforming Complex Points . 60 3.4.1 General Linear Transformations . 60 3.4.2 Reflection . 64 3.4.3 Inversion . 66 3.5 Möbius Transformations . 75 3.6 Cross-Ratio . 85 4 Hyperbolic Geometry 92 4.1 Non-Euclidean Geometry . 92 4.2 The Poincaré Disk Model . 97 4.2.1 Hyperbolic Transformations . 100 4.2.2 Hyperbolic Distance . 115 4.2.3 Geometric Figures . 126 4.3 The Upper Half-Plane Model . 134 5 Triangular Ratio Metric 142 5.1 Definition of the Triangular Ratio Metric . 142 5.2 A Few Examples and Algorithms . 147 5.3 Triangular Ratio Metric Balls . 161 6 Conclusion 167 Alphabetical Index 169 References 172 Appendix 177 R Program 1: The Euler Line of a Complex Triangle . 177 R Program 2: A Möbius Transformation . 180 R Program 3: The Triangular Ratio Metric in a Polygon . 183 R Program 4: The Triangular Ratio Metric in the Unit Disk . 186 R Program 5: The Triangular Ratio Balls with a Contour Map . 189 1 Introduction Considering how long the history of mathematics is, the concept of metric spaces is a quite recent one. The study of it properly began just over a century ago in 1905 when a French mathematician called René Maurice Fréchet (1878-1973) established the first foundations for this new area of mathematics [10],[46]. The first published work about metric spaces was Fréchet’s doctoral dissertation Sur quelques points du calcul fonctionnel, which was submitted in the spring of 1906 [46]. This thesis concerns functional operations and functional calculus that laid the indispensable foundation for the metric spaces but the term "metric space" does actually not occur anywhere in this thesis [46]. Instead the name is due to a German mathematician Felix Hausdorff (1868-1942) [46] who also gave a very important proof related to the converging of Cauchy sequences in complete metric spaces [10]. The study of metric spaces is a sub-area of a larger and older area of mathemat- ics, namely topology. Topology studies the mathematical properties of a geometrical object that are preserved under deformations, twistings, and stretchings but not tearings [69]. Topology can be developed in a metric space [60] but also in a topo- logical space that is a more general concept than metric space [27]. The concepts of both metric spaces and topology in general can be very abstract ones, which is perhaps partly because a more abstract point of view has played a vital role in the modern mathematics of the last century [57]. However, our way of approach to these topics is not an abstract one. Conse- quently, this thesis includes a lot of very detailed proofs for most of the theorems presented and the definitions used are explained in as much detail as possible. While we do introduce all the necessary terms out of which quite a few can be very concep- tual, the very basic geometrical figures such as points, lines and triangles are in our main focus. For instance, we will introduce and prove a formula for a circumcenter of a complex triangle and inspect what happens to straight lines after an inversion in a certain circle. To put it simply, our aim is not only to offer a lot of information about these topics for the reader but also to present the results in a self-contained way that is very easily understandable and does not require a professional level of an ability to read complicated mathematical texts. The contents of this thesis can be divided into four main sections. First, we will explain what a metric space actually is and introduce the definition of a metric. We will give a lot of examples of common metric spaces and also present one that is a little lesser known one. We will give all the necessary concepts related to metric spaces, such as open and closed sets, open balls and fixed points that will be needed later on in this thesis. We will also introduce several different types of continuity that can be generally applied to different functions defined in metric spaces regardless of what kind of a metric space is actually in question. In our next section, we will study complex plane. First, we will have an intro- ductory chapter in which we discuss complex numbers and prove the basic algebraic operations needed when calculating with complex numbers. After that, we will study different geometrical structures from lines to triangles. We will also introduce the concepts of Möbius transformations and cross-ratio which are both very useful not only when inspecting the geometry of the complex plane but also in other later 1 chapters of this thesis. Our third section is about the hyperbolic geometry. The first chapter of also this section is an introduction chapter but instead of complex numbers it will concern the history of the non-Euclidean geometry and give a few necessary definitions such as those of the parallel postulate and geodesics. In the chapter after that we will introduce the Poincaré disk model with its hyperbolic transformations and hyper- bolic distance. At the end of this section, we will also present another model for the hyperbolic geometry, namely the upper half-plane model. The final section of this thesis focuses on another metric, the triangular ratio metric, that is actually very new compared to the ones used in the earlier sections. First, we will define this metric and introduce not only a few example situations but also ways to build an algorithm that calculates the value of this metric. In our last chapter, we will make use of these algorithms and concentrate on the balls created with triangular ratio metric in different domains. In the end of this thesis, there is a conclusion chapter, an alphabetical index, a reference list and an appendix. The index contains not only mathematical terms and concepts introduced but also the names of the famous mathematicians mentioned. While all the sources used can be found in the reference list, the main sources for each chapter are also mentioned in the beginning of the chapter in question. The appendix contains five examples of the R programs used, which are especially useful for the final section about the triangular ratio metric. Writing this thesis has been an interesting and instructive experience. However, while it has required a lot of time and work from me, I am not the only person involved in this process. I would like to thank my supervisor Matti Vuorinen, who suggested the topic of this thesis to me and offered a lot of useful advice during the writing process. Oona Rainio Turku, November 2019 2 2 Metric Spaces In the following chapters, we will introduce the concept of a metric and metric spaces, on which this whole thesis is focused. We will give examples of different metrics and define several essential terms related to subsets of metric spaces. In the final chapter of this section, we will introduce six different types of continuity for functions defined in metric spaces. 2.1 Notion of a Metric In this chapter, we will define a metric and give examples of a few different metrics so that we can understand better what a metric actually is. For most of them, we will also prove that they truly are metrics in the space they are defined in, which shows the common structure of the proof needed for a metric. Mícheál O’Searcord’s book Metric Spaces [48] has been used as the main source for this chapter and this is also where more information about the topic can be found.