The textbook Geometry, published in French by CEDIC/Fernand Nathan andin English by Springer-Verlag (scheduled for 1985) was very favorably received.Nevertheless, many readers found the text too concise and the exercisesat the end of each chapter too difficult, and regretted the absence of any hintsfor the solution of the exercises.This book is intended to respond, at least in part, to these needs. The lengthof the textbook (which will be referred to as [BJ throughout this book) and thevolume of the material covered in it preclude any thought of publishing anexpanded version, but we considered that it might prove both profitable andamusing to some of our readers to have detailed solutions to some of theexercises in the textbook.At the same time, we planned this book to be independent, at least to acertain extent, from the textbook; thus, we have provided summaries of each ofits twenty chapters, condensing in a few pages and under the same titles themost important notions and results used in the solution of the problems. Thestatement of the selected problems follows each summary, and they arenumbered in order, with a reference to the corresponding place in [BJ. Thesereferences are not meant as indications for the solutions of the problems. Inthe body of each summary there are frequent references to FBI, and these canbe helpful in elaborating a point which is discussed too cursorily in this book.Following the summaries we included a number of suggestions and hints forthe solution of the problems; they may well be an intermediate step betweenyour personal solution and ours!The bulk of the book is dedicated to a fairly detailed solution of eachproblem, with references to both this book and the textbook. Following thepractice in [BI, we have made liberal use of illustrations throughout the text.Finally, I would 111cc to express my heartfelt thanks to Springer-Verlag, forincluding this work in their Problem Books in Mathematics series, and to SilvioLevy, for his excellent and speedy translation.Content Level » ResearchRelated subjects » Geometry & TopologyCoverProblems in GeometryCopyright© 1984 by Marcel BergerISBN 0-387-90971-0ISBN 3-540-90971-0QA445.G44513 1984 516'.0076PrefaceContentsChapter 1: Groups Operating on a Set:Nomenclature, Examples, Applications1.A Operation of a Group on a Set ([B, 1.1])1B Transitivity ([B, 1.4])1.C The Erlangen Program: Geometries1.D Stabilizers ([B, 1.5])1.E Orbits; the Class Formula ([B, 1.6])1.F Regular Polyhedra ([B, 1.8])1.G Plane Tilings and Crystallographic Groups ([B, 1.7])1.H More about Tilings; ExercisesProblemsChapter 2: Affine Spaces2.A Affine Spaces; Affine Group ([B, 2.1, 2.3])2.B Affine Maps ([B, 2.3])2.C Homotheties and Dilatations ([B, 2.3.3])2.D Subspaces; Parallelism ([B, 2.4])2.E Independence; Affine Frames ([B, 2.2, 2.4])2.F The Fundamental Theorem of Affine Geometry ([B, 2.6])2.G Finite-dimensional Real Affine Spaces ([B, 2.7])ProblemsChapter 3: Barycenters; the Universal Space3.A Barycenters ([B, 3.4])3.B Associativity of Barycenters ([B, 3.4.8])3.C Barycentric Coordinates ([B, 3.6])3.D A Universal Space ([B, 3.1, 3.2])3.E Polynomials ([B, 3.3])ProblemsChapter 4: Projective Spaces4.A Definition ([B, 4.11)4.B Subspaces, Intersections, Duality ([B,4.6])4.C Homogeneous Coordinates; Charts ([B, 4.2])4.D Projective Bases ([B, 4.4])4.E Morphisms, Homography, Projective Group ([B, 4.5])4.F Perspectives ([B, 4.7])4.G Topology ([B, 4.3])ProblemsChapter 5: Affine-Projective Relationship: Applications5.A The Projective Completion of an Affine Space ([B, 5.1])5.B From Projective to AffineS.C Correspondence between Subspaces ([B, 5.3])5.D Sending Points to Infinity and Back ([B, 5.4J)ProblemsChapter 6: Projective Lines, Cross-Ratios, Homographies6.A Cross-ratios (IB, 6.1, 6.2, 6.3])6.B Harmonic Division ([B, 6.4])6.C Duality ([B, 6.5])6.D Homographies of a Projective Line ([B, 6.6, 6.7])ProblemsChapter 7: Complexifications7.A Complexification of a Vector Space ([B, 7.1, 7.2,7.3, 7.4J)7.B Complexification of a Projective Space ([B, 7.5])7.C Complexification of an Affine Space ([B, 7.6])7.D Adding Up ([B, 7.6])ProblemsChapter 8: More about Euclidean Vector Spaces8.A Definitions ([B, 8.1])8.B Duality and Orthogonality ([B, 8.1])8.C Reflections ([B, 8.2])8.D Structure of O( E) for n = 2 ([B, 8.3])8.E Structure of an Element of O( E) ([B, 8.4J)8.F Angles and Oriented Angles ([B, 8.6, 8.7])8.G Similarities ([B, 8.8])8.H Isotropic Cone, Isotropic Lines, Laguerre Formula ([B, 8.8])8.I Quaternions and Rotations8.J Orientation, Vector Products, Gram Determinants([B, 8.11])ProblemsChapter 9: Euclidean Affine Spaces9.A Definitions ([B, 9.1])9.B Subspaces ([B, 9.2])9.C Structure of an Element of Is( X) ([B, 9.3])9.D Similarities ([B, 9.5])9.E Plane Similarities ([B, 9.6])9.F Metric Properties ([B, 9.7J)9.G Length of Curves ([B, 9.9, 9.10])9.H Canonical Measure, Volumes ([B, 9.12])ProblemsChapter 10: Triangles, Spheres, and Circles10.A Triangles ([B, 10.1, 10.2, 10.3])10.B Spheres ([B, 10.7])10.C Inversion ([B, 10.81)10.D Circles on the Plane and Oriented Angles between Lines ([B, 10.9])ProblemsChapter 11: Convex Sets11.A Definition; First Properties ([B, 11.1, 11.2])11.B The Hahn-Banach Theorem. Supporting Hyperplanes ([B, 11.4, 11.5])11.C Boundary Points of a Convex Set ([B, 11.6])ProblemsChapter 12: Polytopes; Compact Convex Sets12.A Polytopes ([B, 12.1, 12.2, 12.3])12.B Convex Compact Sets ([B, 12.9, 12.10, 12.111)12.C Regular Polytopes ([B, 12.4, 12.5, 12.6])ProblemsChapter 13: Quadratic Forms13.A Definitions ([B, 13.1])13.B Equivalence, Classification ([B, 13.1, 13.4, 13.51)13.C Rank, Degeneracy, Isotropy ([B, 13.21)13.D Orthogonality ([B, 13.31)13.E The Group of a Quadratic Form ([B, 13.6, 13.7])13.F The Two-dimensional Case ([B, 13.8])ProblemsChapter 14: Projective Quadrics14.A Definitions ([B, 14.1])14.B Notation, Examples ([B, 14.1])14.C Classification ([B, 14.1, 14.3, 14.4])14.D Pencils of Quadrics ([B, 14.2])14.E Polarity ([B, 14.5])14.F Duality; Envelope Equation ([B, 14.61)14.G The Group of a Quadric ([B, 14.7])ProblemsChapter 15: Affine Quadrics15.A Definitions ([B, 15.1])15.B Reduction of Affine Quadratic Forms ([B, 15.2, 15.3])15.C Polarity ([B, 15.5])15.D Eucidean Affine Quadrics ([B, 15.6])ProblemsChapter 16: Projective Conics.16.A Notation ([B, 16.1])16.B Good Parametrizations ([B, 16.2])16.C Cross-ratios ([B, 16.2])16.D Homographies of a Conic ([B, 16.3])16.E Intersection of Two Conics; Theorem.of Bezout ([B, 16.4])16.F Pencils of Conics ([B, 16.5])16.G Tangential Conics16.H The Great Poncelet Theorem ([B, 16.6])16.1 Affine Conics ([B, 16.7])ProblemsChapter 17: Euclidean Conics17.A Recapitulation and Notation ([B, 17.11)17.B Foci and Directrices ([B, 17.2])17.C Using the Cyclical Points ([B, 17.4, 17.5])17.D NotesProblemsChapter 18: The Sphere for Its Own Sake18.A Preliminaries ([B, 18.1, 18.2, 18.3])18.B Intrinsic Metric in S ([B, 18.4, 18.5])18.C Spherical Triangles ([B, 18.6])18.D Clifford Parallelism ([B, 10.12, 18.8, 18.9])18.E The Möbius Group ([B, 18.10])ProblemsChapter 19: Elliptic and Hyperbolic Geometry19.A Elliptic Geometry ([B, 19.11)19.B The Hyperbolic Space ([B, 19.2, 19.3})19.C Angles and Trigonometry ([B, 19.2, 19.3])19.D The Conformal Models C and H ([B, 19.6, 19.7])ProblemsChapter 20: The Space of Spheres20.A The Space of Spheres ([B, 20.1])20.B The Canonical Quadratic Form ([B, 10.2])20.C Polysphenc Coordinates ([B, 20.7])ProblemsSuggestions and HintsChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13Chapter 14Chapter 15Chapter 16Chapter 17Chapter 18Chapter 19Chapter 20SolutionsChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13Chapter 14Chapter 15Chapter 16Chapter 17Chapter 18Chapter 19Chapter 20IndexBack Cover