The textbook Geometry, published in French by CEDIC/Fernand Nathan and
in English by Springer-Verlag (scheduled for 1985) was very favorably received.
Nevertheless, many readers found the text too concise and the exercises
at the end of each chapter too difficult, and regretted the absence of any hints
for the solution of the exercises.
This book is intended to respond, at least in part, to these needs. The length
of the textbook (which will be referred to as [BJ throughout this book) and the
volume of the material covered in it preclude any thought of publishing an
expanded version, but we considered that it might prove both profitable and
amusing to some of our readers to have detailed solutions to some of the
exercises in the textbook.
At the same time, we planned this book to be independent, at least to a
certain extent, from the textbook; thus, we have provided summaries of each of
its twenty chapters, condensing in a few pages and under the same titles the
most important notions and results used in the solution of the problems. The
statement of the selected problems follows each summary, and they are
numbered in order, with a reference to the corresponding place in [BJ. These
references are not meant as indications for the solutions of the problems. In
the body of each summary there are frequent references to FBI, and these can
be helpful in elaborating a point which is discussed too cursorily in this book.
Following the summaries we included a number of suggestions and hints for
the solution of the problems; they may well be an intermediate step between
your personal solution and ours!
The bulk of the book is dedicated to a fairly detailed solution of each
problem, with references to both this book and the textbook. Following the
practice in [BI, we have made liberal use of illustrations throughout the text.
Finally, I would 111cc to express my heartfelt thanks to Springer-Verlag, for
including this work in their Problem Books in Mathematics series, and to Silvio
Levy, for his excellent and speedy translation.

Content Level » Research

Related subjects » Geometry & Topology

Cover
Problems in Geometry
Copyright
© 1984 by Marcel Berger
ISBN 0-387-90971-0
ISBN 3-540-90971-0
QA445.G44513 1984 516'.0076
Preface
Contents
Chapter 1: Groups Operating on a Set:Nomenclature, Examples, Applications
1.A Operation of a Group on a Set ([B, 1.1])
1B Transitivity ([B, 1.4])
1.C The Erlangen Program: Geometries
1.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; Exercises
Problems
Chapter 2: Affine Spaces
2.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])
Problems
Chapter 3: Barycenters; the Universal Space
3.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])
Problems
Chapter 4: Projective Spaces
4.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])
Problems
Chapter 5: Affine-Projective Relationship: Applications
5.A The Projective Completion of an Affine Space ([B, 5.1])
5.B From Projective to Affine
S.C Correspondence between Subspaces ([B, 5.3])
5.D Sending Points to Infinity and Back ([B, 5.4J)
Problems
Chapter 6: Projective Lines, Cross-Ratios, Homographies
6.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])
Problems
Chapter 7: Complexifications
7.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])
Problems
Chapter 8: More about Euclidean Vector Spaces
8.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 Rotations
8.J Orientation, Vector Products, Gram Determinants([B, 8.11])
Problems
Chapter 9: Euclidean Affine Spaces
9.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])
Problems
Chapter 10: Triangles, Spheres, and Circles
10.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])
Problems
Chapter 11: Convex Sets
11.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])
Problems
Chapter 12: Polytopes; Compact Convex Sets
12.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])
Problems
Chapter 13: Quadratic Forms
13.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])
Problems
Chapter 14: Projective Quadrics
14.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])
Problems
Chapter 15: Affine Quadrics
15.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])
Problems
Chapter 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 Conics
16.H The Great Poncelet Theorem ([B, 16.6])
16.1 Affine Conics ([B, 16.7])
Problems
Chapter 17: Euclidean Conics
17.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 Notes
Problems
Chapter 18: The Sphere for Its Own Sake
18.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 M?bius Group ([B, 18.10])
Problems
Chapter 19: Elliptic and Hyperbolic Geometry
19.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])
Problems
Chapter 20: The Space of Spheres
20.A The Space of Spheres ([B, 20.1])
20.B The Canonical Quadratic Form ([B, 10.2])
20.C Polysphenc Coordinates ([B, 20.7])
Problems
Suggestions and Hints
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Solutions
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Index
Back Cover