Més enllà de la geometria euclidiana: l'espai hiperbòlic

Abstract

Euclides, around 300 B.C., wrote The Elements, famous work where he brilliantly reflected his conception of plane geometry based on a list of definitions of geometric terms, five logical notions and the following five postulates: 1. A straight line may be drawn joining any two points. 2. A finite straight line may be extended indefinitely in a straight line. 3. A circle may be drawn with any center and any radius. 4. All right angles are equal. 5. If a straight line intersects two straight lines in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The first four postulates clearly have an axiomatic nature, since they are easy to conceive and reflect our intuitive understanding of space around us. However, the fifth postulate seems to be not enough self-evident to be accepted without a proof. In fact, for over two thousand years it was believed that the fifth postulate could be derived from the other four. Carl Friedrich Gauss, in the nineteenth century, was the first to demonstrate that indeed the fifth postulate is independent of the others, and he also made a transcendental discovery: the fifth postulate restricts geometry to a flat universe, without curvature, and by modifying it one obtain new consistent geometries that describe other possible universes with other curvatures, which are called the non-euclidian geometries. Unfortunately, Gauss never published this results. Years later, noneuclidian geometries were rediscovered independently by Nikolai Lobachevsky and J ́anos Bolyai. The historical papers, where these ideas were published for the first time, can be consulted: On the principles of geometry, Lobachevsky (1829) and The absolute science of space, Bolyai (1832). Moreover, the modern geometric tools developed by Behrnard Riemann and Gauss himself among others, make use of the differential analysis in order to describe the space in a less axiomatic and more universal way, so that the different possible geometries have emerged naturally and perfectly classified and have became useful to develop physical theories with profound implications as the Theory of General Relativity of Einstein. The main goal of this work is to show how modern geometry, the Riemannian, has allowed to classify the different spaces in terms of their curvatures and to establish dualities between them. This development is carried out in the first chapter, where we demonstrate the fundamental results that solve the problem. As we shall see, by imposing that the curvature is constant, we reach to three possible spaces: the euclidian one (non curved), the spherical one (positively curved) and the hyperbolic one (negatively curved). The first two cases are briefly described in the first chapter.Additionally, we show how the isometries of the space can be used to identify the manifolds that contains. The most surprising result that we obtain, without any doubt, is the existence of the hyperbolic space, which is totally consistent by itself and presents clear dualities with the other two. Furthermore, it reflects the space-time geometry according to the Theory of General Relativity. Hence, it is the right one to capture the metrical properties of the Universe. However, the hyperbolic space can not be intuitively conceived and has an elusive nature. This is the motivation of the second part of the work that is contained in the second and third chapters. The aim is to familiarize the reader with this space for an arbitrary dimension through different models that reduce the level of abstraction that it presents. Therefore, the different models are presented in a constructive way in the second chapter by using the tools of the Riemannian geometry. In the third chapter, the different isometric transformations of the hyperbolic space are shown, allowing the study of its invariances and its manifolds. In order to present a self-contained text, we include an appendix where all the essential ideas of Riemannian geometry that we use in the work are developed. This extra chapter also contains some appreciations that motivate the objectives of the work. Several sources of information have been used with the intention of optimizing the amount of formalism and results presented. In order to facilitate the deepening of the contents by the reader, at the beginning of each chapter we indicate the different sources that can be followed in parallel with the text.