DOCSLIB.ORG

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hyperbolic Geometry: Isometry Groups of Hyperbolic Space HYPERBOLIC GEOMETRY: ISOMETRY GROUPS OF HYPERBOLIC SPACE ILYAS KHAN Abstract. The goal of this paper is twofold. First, it consists of an intro- duction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that pre- serve orientation, and deals with the topological properties of their discrete subgroups. In the final section, we show that if one of these Fuchsian groups satisfies certain conditions, then its quotient space is a 2-manifold for which the hyperbolic plane is a covering space. Contents 1. Hyperbolic Geometry and PSL(2,R) 1 2. Geodesics 5 3. Discrete Isometry Groups and Proper Discontinuity 8 4. Topological Properties of Fuchsian Groups 12 Acknowledgments 15 References 16 1. Hyperbolic Geometry and PSL(2,R) There are several models of hyperbolic space, but for the purposes of this paper we will restrict our view to the Lobachevski upper half plane. Definition 1.1. The upper half plane is the subset H = fz 2 C : Im(z) > 0g of C with the metric pdx2 + dy2 ds = y where z = x + iy. We can use this metric to find the length of a differentiable path γ through hyperbolic space by integrating over its domain. The hyperbolic length, h(γ) of the differentiable path γ : [0; 1] ! H where γ(t) = (x(t); y(t)) is q 1 dx 2 dy 2 1 dz Z ( dt ) + ( dt ) Z j j h(γ) = dt = dt dt 0 y(t) 0 y(t) q dx 2 dy 2 dz since ( dt ) + ( dt ) is the Euclidean norm of the vector dt . Date: August 24, 2012. 1 2 ILYAS KHAN Definition 1.2. The hyperbolic distance ρ(z; w) between z; w 2 H is given by ρ(z; w) = inf h(γ) where γ is an element of the set of all differentiable paths connecting z and w. Definition 1.3. An isometry of H is a function g : H ! H such that for any z; w 2 H, ρ(z; w) = ρ(g(z); g(w)). This paper will focus on the properties of special groups of isometries of H. Definition 1.4. A group is a set G with an operation ∗ which satisfies the axioms G1 through G3 (G1) ∗ is associative. (G2) There is an element e in G such that a ∗ e = a and e ∗ a = a for every element a in G. (G3) For every element a in G, there is an element a−1 in G such that a∗a−1 = e and a−1 ∗ a = e. One group of isometries (of R2) particularly important to our aims is the special linear group SL(2,R). SL(2,R) is the set of 2 by 2 matrices with determinant 1 under the group operation of matrix multiplication. That SL(2,R) is a group is easy to show using the multiplicity of determinants, the existence of an identity matrix, and the inversion formula for 2 by 2 matrices. Definition 1.5. The elements of the projective special linear group PSL(2,R) are az+b the M¨obiustransformations, rational functions from C to C of the form z 7! cz+d with ad − bc = 1. We now want to show that PSL(2,R) is a group. We see immediately that the set az+b PSL(2,R) is isomorphic to SL(2,R) modulo {±Idg, because for any element cz+d a factor of −1 in the numerator will cancel with a factor of −1 in the denominator. Hence, we set up a surjective identification a b az + b g( ) = c d cz + d from SL(2,R) to PSL(2,R). Definition 1.6. If G and H are groups, a homomorphism from G to H is a function f : G ! H such that for any two elements a and b in G f(ab) = f(a)f(b): Let the group operation on PSL(2,R) be function composition. A simple calcu- lation confirms that composition of two M¨obiustransformations corresponds to the multiplication of the corresponding 2 by 2 matrices in SL(2,R). Furthermore, the inverse matrices and identity matrices of SL(2,R) on the subset SL(2,R) ={±Idg correspond to the inverse and identity elements of PSL(2,R). We conclude that PSL(2,R) is a group and that our identification g is a surjective group homomor- phism. Hence, we can say that PSL(2,R) is a homomorphic image of SL(2,R). PSL(2,R) plays an important part in our overall goal to classify the isometries of H. We must first prove that the elements of PSL(2,R) are in fact isometries, and then we will examine what properties distinguish PSL(2,R) from the rest of the isometries of H. HYPERBOLIC GEOMETRY: ISOMETRY GROUPS OF HYPERBOLIC SPACE 3 Lemma 1.7. Every M¨obiustransformation φ in PSL(2,R) is a homeomorphism of H. az+b Proof. First show that φ maps H into H. Let φ(z) = cz+d where ad − bc = 1 and z 2 H. (az + b)(cz + d) acjzj2 + adz + bcz + bd φ(z) = = (cz + d)(cz + d) jcz + dj2 φ(z) − φ(z) adz + bcz − adz − bcz Im(φ(z)) = = 2i 2ijcz + dj2 (ad − bc)(z − z) Im(z) = = 2ijcz + dj2 jcd + zj2 The imaginary component of z is greater than 0, so clearly Im(φ(z)) > 0, which implies that φ(z) is in H. Because φ is a rational function with a nonzero denominator, continuity is clear. The existence of an inverse follows from the fact that φ−1 is also an element of PSL(2,R). By the argument above, it is continuous and maps into H. Therefore, φ is a homeomorphism of H. Theorem 1.8. PSL(2,R) ⊂ Isom(H). Proof. Let γ : [0; 1] ! H be a piecewise differentiable path in H. Let γ be given by z(t) = (x(t); y(t)) and w(t) = T (z(t)) = u(t) + iv(t) By the quotient rule, dw a(cz + d) − c(az + b) acz − caz + ad − bc 1 = = = dt (cz + d)2 (cz + d)2 (cz + d)2 Im(z) dw v Since Im(T (z)) = jcz+dj2 = v and y = Im(T (z)) by Lemma 1.7, j dt j = y . From the definition of hyperbolic length, we obtain the following equation. v(t) Z 1 j dw j Z 1 j dw dz j Z 1 j dz j Z 1 j dz j h(T (γ)) = dt dt = dz dt dt = y(t) dt dt = dt dt = h(γ): 0 v(t) 0 v(t) 0 v(t) 0 y(t) The hyperbolic distance ρ(j; k) is the infimum of the differentiable paths γ be- tween j and k, so since each γ is invariant under T, ρ(j; k) is invariant under T. Thus, T 2 PSL(2,R) is an isometry. [1] There is one further property that classifies PSL(2,R) as a subset of Isom(H), which is that PSL(2,R) is the set of all orientation preserving isometries of H. A linear operator on a vector space is orientation preserving if its determinant is positive. The equivalent notion for M¨obiustransformations is the following: Definition 1.9. Let φ be a map H ! H. Then φ is orientation preserving if det(Dφz) > 0 for all z 2 H. Definition 1.10. The tangent space at z in H, written TzH, is the vector space generated by the collection of vectors based at z tangent to a curve through z. The derivative of φ at z Dφz is a linear map from TzH to Tφ(z)H. To show that PSL(2,R) is orientation preserving, we will establish an inner product on TzH. Ob- serve that each point in H has two linearly independent tangent vectors associated 2 ∼ 2 with it and that H is a two dimensional subset of R , which imply that TzH = R . This observation motivates the following definition. 4 ILYAS KHAN Definition 1.11. The inner product of v; w 2 TzH is given by 1 < v; w > = < v; w > 2 z (Im(z))2 R 2 2 where < v; w >R is the standard dot product for R . Definition 1.12. A function φ : H ! H is an isometry if for all z 2 H and v; w 2 TzH < v; w >z=< Dφz(v); Dφz(w) >φ(z) : Remark 1.13. It follows from a simple calculation that this definition is equivalent to Definition 1.3. Lemma 1.14. If σ and φ preserve orientation, then φ ◦ σ preserves orientation. Proof. This follows immediately from the multiplicity of determinants. D(φ◦σ)z = Dφσ(z) ◦ Dσz by the chain rule. Then, det D(φ ◦ σ)z = det(Dφσ(z) ◦ Dσz) = det Dφσ(z) · det Dσz > 0, since det Dφσ(z) > 0 and det Dσz > 0 since φ and σ are orientation preserving. Lemma 1.15. Any isometry of H of the form z 7! yz + x where y; x 2 R; y 6= 0 and z 2 H preserves orientation. Proof. Let z = r + is, or equivalently z = (r; s) 2 C where r; s 2 R. Let f(r; s) = (yr + x; ys). @f1 @f1 y 0 Df = @r @s = (r;s) @f2 @f2 0 y @r @s 2 det(Df(r;s)) = y > 0, so f preserves orientation. Theorem 1.16.
Load more

Recommended publications
  • Geometric Manifolds
    Wintersemester 2015/2016 University of Heidelberg Geometric Structures on Manifolds Geometric Manifolds by Stephan Schmitt Contents Introduction, first Definitions and Results 1 Manifolds - The Group way .................................... 1 Geometric Structures ........................................ 2 The Developing Map and Completeness 4 An introductory discussion of the torus ............................. 4 Definition of the Developing map ................................. 6 Developing map and Manifolds, Completeness 10 Developing Manifolds ....................................... 10 some completeness results ..................................... 10 Some selected results 11 Discrete Groups .......................................... 11 Stephan Schmitt INTRODUCTION, FIRST DEFINITIONS AND RESULTS Introduction, first Definitions and Results Manifolds - The Group way The keystone of working mathematically in Differential Geometry, is the basic notion of a Manifold, when we usually talk about Manifolds we mean a Topological Space that, at least locally, looks just like Euclidean Space. The usual formalization of that Concept is well known, we take charts to ’map out’ the Manifold, in this paper, for sake of Convenience we will take a slightly different approach to formalize the Concept of ’locally euclidean’, to formulate it, we need some tools, let us introduce them now: Definition 1.1. Pseudogroups A pseudogroup on a topological space X is a set G of homeomorphisms between open sets of X satisfying the following conditions: • The Domains of the elements g 2 G cover X • The restriction of an element g 2 G to any open set contained in its Domain is also in G. • The Composition g1 ◦ g2 of two elements of G, when defined, is in G • The inverse of an Element of G is in G. • The property of being in G is local, that is, if g : U ! V is a homeomorphism between open sets of X and U is covered by open sets Uα such that each restriction gjUα is in G, then g 2 G Definition 1.2.
    [Show full text]
  • Black Hole Physics in Globally Hyperbolic Space-Times
    Pram[n,a, Vol. 18, No. $, May 1982, pp. 385-396. O Printed in India. Black hole physics in globally hyperbolic space-times P S JOSHI and J V NARLIKAR Tata Institute of Fundamental Research, Bombay 400 005, India MS received 13 July 1981; revised 16 February 1982 Abstract. The usual definition of a black hole is modified to make it applicable in a globallyhyperbolic space-time. It is shown that in a closed globallyhyperbolic universe the surface area of a black hole must eventuallydecrease. The implications of this breakdown of the black hole area theorem are discussed in the context of thermodynamics and cosmology. A modifieddefinition of surface gravity is also given for non-stationaryuniverses. The limitations of these concepts are illustrated by the explicit example of the Kerr-Vaidya metric. Keywocds. Black holes; general relativity; cosmology, 1. Introduction The basic laws of black hole physics are formulated in asymptotically flat space- times. The cosmological considerations on the other hand lead one to believe that the universe may not be asymptotically fiat. A realistic discussion of black hole physics must not therefore depend critically on the assumption of an asymptotically flat space-time. Rather it should take account of the global properties found in most of the widely discussed cosmological models like the Friedmann models or the more general Robertson-Walker space times. Global hyperbolicity is one such important property shared by the above cosmo- logical models. This property is essentially a precise formulation of classical deter- minism in a space-time and it removes several physically unreasonable pathological space-times from a discussion of what the large scale structure of the universe should be like (Penrose 1972).
    [Show full text]
  • Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances
    Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent.
    [Show full text]
  • On Isometries on Some Banach Spaces: Part I
    Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections On isometries on some Banach spaces { something old, something new, something borrowed, something blue, Part I Dijana Iliˇsevi´c University of Zagreb, Croatia Recent Trends in Operator Theory and Applications Memphis, TN, USA, May 3{5, 2018 Recent work of D.I. has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Something old, something new, something borrowed, something blue is referred to the collection of items that helps to guarantee fertility and prosperity. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. A linear map ': X!Y is called a linear isometry if k'(x)k = jxj; x 2 X : We shall be interested in surjective linear isometries on Banach spaces. One of the main problems is to give explicit description of isometries on a particular space. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field.
    [Show full text]
  • Computing the Complexity of the Relation of Isometry Between Separable Banach Spaces
    Computing the complexity of the relation of isometry between separable Banach spaces Julien Melleray Abstract We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [1] and Mayer-Wolf [4]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. 1 Introduction Over the past fteen years or so, the theory of complexity of Borel equivalence relations has been a very active eld of research; in this paper, we compute the complexity of a relation of geometric nature, the relation of (linear) isom- etry between separable Banach spaces. Before stating precisely our result, we begin by quickly recalling the basic facts and denitions that we need in the following of the article; we refer the reader to [3] for a thorough introduction to the concepts and methods of descriptive set theory. Before going on with the proof and denitions, it is worth pointing out that this article mostly consists in putting together various results which were already known, and deduce from them after an easy manipulation the complexity of the relation of isometry between separable Banach spaces. Since this problem has been open for a rather long time, it still seems worth it to explain how this can be done, and give pointers to the literature. Still, it seems to the author that this will be mostly of interest to people with a knowledge of descriptive set theory, so below we don't recall descriptive set-theoretic notions but explain quickly what Lipschitz-free Banach spaces are and how that theory works.
    [Show full text]
  • Hyperbolic Geometry, Fuchsian Groups, and Tiling Spaces
    HYPERBOLIC GEOMETRY, FUCHSIAN GROUPS, AND TILING SPACES C. OLIVARES Abstract. Expository paper on hyperbolic geometry and Fuchsian groups. Exposition is based on utilizing the group action of hyperbolic isometries to discover facts about the space across models. Fuchsian groups are character- ized, and applied to construct tilings of hyperbolic space. Contents 1. Introduction1 2. The Origin of Hyperbolic Geometry2 3. The Poincar´eDisk Model of 2D Hyperbolic Space3 3.1. Length in the Poincar´eDisk4 3.2. Groups of Isometries in Hyperbolic Space5 3.3. Hyperbolic Geodesics6 4. The Half-Plane Model of Hyperbolic Space 10 5. The Classification of Hyperbolic Isometries 14 5.1. The Three Classes of Isometries 15 5.2. Hyperbolic Elements 17 5.3. Parabolic Elements 18 5.4. Elliptic Elements 18 6. Fuchsian Groups 19 6.1. Defining Fuchsian Groups 20 6.2. Fuchsian Group Action 20 Acknowledgements 26 References 26 1. Introduction One of the goals of geometry is to characterize what a space \looks like." This task is difficult because a space is just a non-empty set endowed with an axiomatic structure|a list of rules for its points to follow. But, a priori, there is nothing that a list of axioms tells us about the curvature of a space, or what a circle, triangle, or other classical shapes would look like in a space. Moreover, the ways geometry manifests itself vary wildly according to changes in the axioms. As an example, consider the following: There are two spaces, with two different rules. In one space, planets are round, and in the other, planets are flat.
    [Show full text]
  • Isometries and the Plane
    Chapter 1 Isometries of the Plane \For geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. (W. K. Clifford) The focus of this first chapter is the 2-dimensional real plane R2, in which a point P can be described by its coordinates: 2 P 2 R ;P = (x; y); x 2 R; y 2 R: Alternatively, we can describe P as a complex number by writing P = (x; y) = x + iy 2 C: 2 The plane R comes with a usual distance. If P1 = (x1; y1);P2 = (x2; y2) 2 R2 are two points in the plane, then p 2 2 d(P1;P2) = (x2 − x1) + (y2 − y1) : Note that this is consistent withp the complex notation. For P = x + iy 2 C, p 2 2 recall that jP j = x + y = P P , thus for two complex points P1 = x1 + iy1;P2 = x2 + iy2 2 C, we have q d(P1;P2) = jP2 − P1j = (P2 − P1)(P2 − P1) p 2 2 = j(x2 − x1) + i(y2 − y1)j = (x2 − x1) + (y2 − y1) ; where ( ) denotes the complex conjugation, i.e. x + iy = x − iy. We are now interested in planar transformations (that is, maps from R2 to R2) that preserve distances. 1 2 CHAPTER 1. ISOMETRIES OF THE PLANE Points in the Plane • A point P in the plane is a pair of real numbers P=(x,y). d(0,P)2 = x2+y2. • A point P=(x,y) in the plane can be seen as a complex number x+iy.
    [Show full text]
  • Isometry and Automorphisms of Constant Dimension Codes
    Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Constant Dimension Codes Anna-Lena Trautmann Institute of Mathematics University of Zurich \Crypto and Coding" Z¨urich, March 12th 2012 1 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 2 / 25 isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding 3 / 25 automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes 3 / 25 automorphism groups are canonical representative of orbit codes Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding 3 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes 3 / 25 Definition Subspace metric: dS(U; V) = dim(U + V) − dim(U\V) Injection metric: dI (U; V) = max(dim U; dim V) − dim(U\V) Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes Definition n n The projective geometry P(Fq ) is the set of all subspaces of Fq .
    [Show full text]
  • Geodesic Planes in Hyperbolic 3-Manifolds
    Geodesic planes in hyperbolic 3-manifolds Curtis T. McMullen, Amir Mohammadi and Hee Oh 28 April 2015 Abstract This paper initiates the study of rigidity for immersed, totally geodesic planes in hyperbolic 3-manifolds M of infinite volume. In the case of an acylindrical 3-manifold whose convex core has totally geodesic boundary, we show that the closure of any geodesic plane is a properly immersed submanifold of M. On the other hand, we show that rigidity fails for quasifuchsian manifolds. Contents 1 Introduction . 1 2 Background . 8 3 Fuchsian groups . 10 4 Zariski dense Kleinian groups . 12 5 Influence of a Fuchsian subgroup . 16 6 Convex cocompact Kleinian groups . 17 7 Acylindrical manifolds . 19 8 Unipotent blowup . 21 9 No exotic circles . 25 10 Thin Cantor sets . 31 11 Isolation and finiteness . 31 A Appendix: Quasifuchsian groups . 34 B Appendix: Bounded geodesic planes . 36 Research supported in part by the Alfred P. Sloan Foundation (A.M.) and the NSF. Typeset 2020-09-23 11:37. 1 Introduction 3 Let M = ΓnH be an oriented complete hyperbolic 3-manifold, presented as the quotient of hyperbolic space by the action of a Kleinian group ∼ + 3 Γ ⊂ G = PGL2(C) = Isom (H ): Let 2 f : H ! M be a geodesic plane, i.e. a totally geodesic immersion of a hyperbolic plane into M. 2 We often identify a geodesic plane with its image, P = f(H ). One can regard P as a 2{dimensional version of a Riemannian geodesic. It is natural to ask what the possibilities are for its closure, V = f(H2) ⊂ M: When M has finite volume, Shah and Ratner showed that strong rigidity properties hold: either V = M or V is a properly immersed surface of finite area [Sh], [Rn2].
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
    [Show full text]
  • Hyperbolic Geometry, Surfaces, and 3-Manifolds Bruno Martelli
    Hyperbolic geometry, surfaces, and 3-manifolds Bruno Martelli Dipartimento di Matematica \Tonelli", Largo Pontecorvo 5, 56127 Pisa, Italy E-mail address: martelli at dm dot unipi dot it version: december 18, 2013 Contents Introduction 1 Copyright notices 1 Chapter 1. Preliminaries 3 1. Differential topology 3 1.1. Differentiable manifolds 3 1.2. Tangent space 4 1.3. Differentiable submanifolds 6 1.4. Fiber bundles 6 1.5. Tangent and normal bundle 6 1.6. Immersion and embedding 7 1.7. Homotopy and isotopy 7 1.8. Tubolar neighborhood 7 1.9. Manifolds with boundary 8 1.10. Cut and paste 8 1.11. Transversality 9 2. Riemannian geometry 9 2.1. Metric tensor 9 2.2. Distance, geodesics, volume. 10 2.3. Exponential map 11 2.4. Injectivity radius 12 2.5. Completeness 13 2.6. Curvature 13 2.7. Isometries 15 2.8. Isometry group 15 2.9. Riemannian manifolds with boundary 16 2.10. Local isometries 16 3. Measure theory 17 3.1. Borel measure 17 3.2. Topology on the measure space 18 3.3. Lie groups 19 3.4. Haar measures 19 4. Algebraic topology 19 4.1. Group actions 19 4.2. Coverings 20 4.3. Discrete groups of isometries 20 v vi CONTENTS 4.4. Cell complexes 21 4.5. Aspherical cell-complexes 22 Chapter 2. Hyperbolic space 25 1. The models of hyperbolic space 25 1.1. Hyperboloid 25 1.2. Isometries of the hyperboloid 26 1.3. Subspaces 27 1.4. The Poincar´edisc 29 1.5. The half-space model 31 1.6.
    [Show full text]
  • Hyperbolic Manifolds Hilary Term 2000 Marc Lackenby
    Hyperbolic Manifolds Hilary Term 2000 Marc Lackenby Geometry and topology is, more often than not, the study of manifolds. These manifolds come in a variety of different flavours: smooth manifolds, topological manifolds, and so on, and many will have extra structure, like complex manifolds or symplectic manifolds. All of these concepts can be brought together into one overall definition. A pseudogroup on a (topological) manifold X is a set G of homeomorphisms between open subsets of X satisfying the following conditions: 1. The domains of the elements of G must cover X. 2. The restriction of any element of G to any open set in its domain is also in G. 3. The composition of two elements of G, when defined, is also in G. 4. The inverse of an element of G is in G. 5. The property of being in G is ‘local’, that is, if g: U → V is a homeomorphism between open sets of X, and U has a cover by open sets Uα such that g|Uα is in G for each Uα, then g is in G. For example, the set of all diffeomorphisms between open sets of Rn forms a pseudogroup. A G-manifold is a Hausdorff topological space M with a countable G-atlas. A G-atlas is a collection of G-compatible co-ordinate charts whose domains cover M. A co-ordinate chart is a pair (Ui, φi), where Ui is an open set in M and φi: Ui → X is a homeomorphism onto its image. That these are G-compatible means that −1 whenever (Ui, φi) and (Uj, φj) intersect, the transition map φi ◦φj : φj(Ui ∩Uj) → φi(Ui ∩ Uj ) is in the pseudogroup G.
    [Show full text]