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Geometry and Topology Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics. This book introduces the ideas of geometry and includes a generous supply of simple explanations and examples. The treatment emphasizes coordinate systems and the coordinate changes that generate symmetries. The discussion moves from Euclidean to non-Euclidean geometries, including spherical and hyperbolic geometry, and then on to affine and projective linear geometries. Group theory is introduced to treat geometric symmetries, leading to the unification of geometry and group theory in the Erlangen program. An introduction to basic topology follows, with the Möbius strip, the Klein bottle and the surface with g handles exemplifying quotient topologies and the homeomorphism problem. Topology combines with group theory to yield the geometry of transformation groups, having applications to relativity theory and quantum mechanics. A final chapter features historical discussions and indications for further reading. While the book requires minimal prerequisites, it provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. The text is based on many years' teaching experience and is thoroughly class-tested. There are copious illustrations, and each chapter ends with a wide supply of exercises. Further teaching material is available for teachers via the web, including assignable problem sheets with solutions. Miles Reid Mathematics Institute, University of Warwick Bálazs Szendroi Mathematical Institute, University of Oxford, and Martin Powell Fellow in Pure Mathematics at St Peter’s College, Oxford Cambridge University Press Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org 978-0-521-84889-3 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 ISBN-13: 978-0-511-13733-4 (eBook - NetLibrary) ISBN-10: 0-511-13733-8 (eBook - NetLibrary) ISBN-13: 978-0-521-84889-3 (hardback) ISBN-10: 0-521-84889-X (hardback) ISBN-13: 978-0-521-61325-5 (paperback) ISBN-10: 0-521-61325-6 (paperback) Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents List of figures Preface 1 Euclidean geometry 1.1 The metric on Rn 1.2 Lines and collinearity in Rn 1.3 Euclidean space En 1.4 Digression: shortest distance 1.5 Angles 1.6 Motions 1.7 Motions and collinearity 1.8 A motion is affine linear on lines 1.9 Motions are affine transformations 1.10 Euclidean motions and orthogonal transformations 1.11 Normal form of an orthogonal matrix 1.11.1 The 2 × 2 rotation and reflection matrices 1.11.2 The general case 1.12 Euclidean frames and motions 1.13 Frames and motions of E2 1.14 Every motion of E2 is a translation, rotation, reflection or glide 1.15 Classification of motions of E3 1.16 Sample theorems of Euclidean geometry 1.16.1 Pons asinorum 1.16.2 The angle sum of triangles 1.16.3 Parallel lines and similar triangles 1.16.4 Four centres of a triangle 1.16.5 The Feuerbach 9-point circle Exercises 2 Composing maps 2.1 Composition is the basic operation 2.2 Composition of affine linear maps x → Ax + b 3 Spherical and hyperbolic non-Euclidean geometry 3.1 Basic definitions of spherical geometry 3.2 Spherical triangles and trigonometry 3.3 The spherical triangle inequality 3.4 Spherical motions 3.5 Properties of S2 like E2 3.6 Properties of S2 unlike E2 3.7 Preview of hyperbolic geometry 3.8 Hyperbolic space 3.9 Hyperbolic distance 3.10 Hyperbolic triangles and trigonometry 3.11 Hyperbolic motions 3.12 Incidence of two lines in H2 3.13 The hyperbolic plane is non-Euclidean 3.14 Angular defect 3.14.1 The first proof 3.14.2 An explicit integral 3.14.3 Proof by subdivision 3.14.4 An alternative sketch proof Exercises 4 Affine geometry 4.1 Motivation for affine space 4.2 Basic properties of affine space 4.3 The geometry of affine linear subspaces 4.4 Dimension of intersection 4.5 Affine transformations 4.6 Affine frames and affine transformations 4.7 The centroid Exercises 5 Projective geometry 5.1 Motivation for projective geometry 5.1.1 Inhomogeneous to homogeneous 5.1.2 Perspective 5.1.3 Asymptotes 5.1.4 Compactification 5.2 Definition of projective space 5.3 Projective linear subspaces 5.4 Dimension of intersection 5.5 Projective linear transformations and projective frames of reference 5.6 Projective linear maps of P1 and the cross-ratio 5.7 Perspectivities 5.8 Affine space An as a subset of projective space Pn 5.9 Desargues’ theorem 5.10 Pappus’ theorem 5.11 Principle of duality 5.12 Axiomatic projective geometry Exercises 6 Geometry and group theory 6.1 Transformations form a group 6.2 Transformation groups 6.3 Klein’s Erlangen program 6.4 Conjugacy in transformation groups 6.5 Applications of conjugacy Exercises Appendix B: Lorentzian and Hermitian geometry B.1 Orthonormal bases f1, ..., fn ∈ Rn If (V, ?) is Lorentzian, a matrix A satisfying the condition tA J A = J of Proposition B.4(3) is called a Lorentz matrix. This section discusses a slight variant of the above material for vector spaces over the field C of complex numbers. Let V be a finite-dimensional vector space over C. A Hermitian form ? : V × V → C is a map satisfying the conditions ?(?u + µv, w) = ??(u, w) + µ?(v, w), and ?(u, ?v + µw) = ??(u, v) + µ?(u, w), where ?, µ ∈ C; note the appearance of the complex conjugate in the first row. The corresponding Hermitian norm q on V is given by q(v) = |?(v, v)|. The relation between ? and q is slightly more complicated than in the real case; I leave you to check the rather daunting looking identity ?(u, v) = 1/2 [q(u + v) - q(u - v) + i(q(u + iv) - q(u - iv))]. The terms in the identity are not so important; what is important is the fact that q gives back ?. Since I am only interested in a special case, I choose a basis {e1, ..., en} of V straight away and assume that ?(?1e1 + · · · + ?nen, µ1e1 + · · · + µnen) = ?1µ1 + · · · + ?nµn. Such a form is called a definite Hermitian form. Under ?, e1, ..., en form an orthonormal basis: ?(ei , ej ) = δij . The following is completely analogous to Proposition B.4. Proposition Let ? : V → V be a linear map represented by the n × n matrix A in the given basis. Then the following are equivalent: 1. ? preserves the norm q; 2. ? preserves the Hermitian form ?; 3. A satisfies hA tA = In, where hA is the Hermitian conjugate defined by hA = tA; that is, (hA)ij = Aji. The transformation ? or the matrix A representing it is unitary if it satisfies these conditions; the set of n × n unitary matrices is denoted U(n). Unitary transformations (possibly on infinite-dimensional spaces) have many pleasant properties which makes them ubiquitous in mathematics. They are also the basic building blocks of quantum mechanics and hence presumably nature; in this book I discuss one tiny example of this in 8.7. Exercises B.1 Let f1 = (2/3, 1/3, 2/3) and f2 = (1/3, 2/3, -2/3) ∈ R3; find all vectors f3 ∈ R3 for which f1, f2, f3 is an orthonormal basis. B.2 By writing down explicitly the conditions for a 2 × 2 matrix to be Lorentzian, show that any such matrix has the form cosh s sinh s or cosh s - sinh s. sinh s cosh s sinh s - cosh s B.3 This exercise is a generalization of the previous one; it shows that any Lorentzian matrix can be put in a simple normal form in a suitable Lorentz basis; the Euclidean case is included in the main text in 1.11. Let ? : Rn+1 → Rn+1 be a linear map given by a Lorentzian matrix A. Prove that there exists a Lorentz basis of Rn+1 in which the matrix of ? is uf8eb ±1 uf8f6 uf8eb B0 uf8f6 B = uf8ec uf8ec uf8ec uf8ec uf8ed Ik+ ?Ik? B1 . uf8f7 uf8f7 uf8f7 uf8f7 uf8f8 or B = uf8ec uf8ec uf8ec uf8ec uf8ed Ik+ ?Ik? B1 . uf8f7 uf8f7 uf8f7 uf8f7 uf8f8 Bl Bl-1 where B0 = ± cosh ?0 sinh ?0 sinh ?0 cosh ?0 , Bi = cos ?i - sin ?i sin ?i cos ?i for i > 0, and Ik± are identity matrices. _Hint: argue as in the Euclidean case in 1.11.2; the only extra complication is that you have to take into account the sign of the Lorentz form on the eigenvectors._ The statement follows by sorting out the cases that can arise. References [1] Michael Artin, Algebra, Englewood Cliffs, NJ: Prentice Hall, 1991. [2] Alan F. Beardon, The Geometry of Discrete Groups, New York: Springer, 1983. [3] Roberto Bonola, Non-Euclidean Geometry: A Critical and Historical Study of its Developments, New York: Dover, 1955. [4] J. H. Conway and D. A. 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