Marcel Berger
Geometry I
Translated from the French by M. Cole and S. Levy
A
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fyj Springer
Table of contents
Volume I
Introduction
Chapter 0 . N o t a t i o n and background
0.1 Set theory
0 I Algebra
0.3 Metric spaces
0.4 General topology
0.5 Hyperbolic trigonometry
0.6 Lebesgue measure; integration theory
xi
1
1
1
2
2
3
3
Chapter 1. G r o u p actions: examples and applications . . . .
1 1 Group actions
. . .
1.2 Examples
1.3 Faithful actions
1 4 Transitive actions
1.5 Stabilizers; homogeneous spaces
1.6 Orbits; the class formula
1.7 Tilings and crystallographic groups
'.
1.8 Tilings of the two-sphere and regular polyhedra
1.9 Exercises
4
5
5
6
6
7
9
11
22
28
Chapter 2 . Affine spaces
2.1 Definitions
2.2 Examples. Affine frames
2.3 Affine morphisms
2.4 Affine subspaces
2.5 Geometry at last: Thales, Pappus, Desargues
2.6 The fundamental theorem of affine geometry
2.7 Finite-dimensional real affine spaces
32
33
34
37
42
49
52
56
VI
Table of c o n t e n t s
2.8 Exercises
C h a p t e r 3. Barycenters; the universal space
3.1 The universal space
3.2 Morphisms and the universal space
3.3 Polynomials over an affine space
3.4 Barycenters
3.5 Relationship with affine maps and subspaces
3.6 Barycentric coordinates
3.7 Exercises
64
.
67
68
71
72
75
79
81
83
C h a p t e r 4. Projective spaces
85
4.0 Introduction
86
4.1 Definition and examples
87
4.2 Structure of projective spaces: charts
88
4.3 Structure of projective spaces: topology and algebraic topology 90
4.4 Projective bases
95
4.5 Morphisms
96
4.6 Subspaces
100
4.7 Perspective; aerial photography
102
4.8 The non-commutative case
105
4.9 Exercises
109
C h a p t e r 5. Affine-projective relationship: a p p l i c a t i o n s . . . .
5.0 Introduction
5.1 The projective completion of an affine space . •
5.2 Examples
5.3 Relationship between affine and projective subspaces. Parallelism
5.4 Sending objects to infinity; applications
5.5 Exercises
C h a p t e r 6. Projective lines, cross-ratios, homographies
6.1 Definition of cross-ratios
6.2 Computation of cross-ratios
6.3 Action of permutations
6.4 Harmonic division
*
6.5 Cross-ratios and duality; applications
6.6 Homographies of a projective line
6.7 Involutions
6.8 Exercises
C h a p t e r 7. Complexifications
7.0 Introduction . . .
7.1 Complexification of a real vector space
Ill
112
113
114
115
116
119
. . . 122
123
124
126
127
130
134
137
138
142
143
145
Table of contents
7.2
7.3
7.4
7.5
7.6
7.7
Functoriality; complexification of morphisms
Complexification of polynomials
Subspaces and complexifications
Complexification of a projective space
Complexification of an affine space
Exercises
vii
146
146
147
148
148
150
C h a p t e r 8. Euclidean vector spaces
151
8.1 Definition and basic properties
153
8.2 The orthogonal group: first properties and plan of attack . . 157
8.3 The two-dimensional case
161
8.4 Structure of elements of O(E). Generators for O(E) and
O+(E)
165
8.5 Simplicity of O(E)
168
8.6 Angles between lines and half-lines
171
8.7 Oriented angles in the plane
173
8.8 Similarities. Isotropic cone and lines
183
8.9 Quaternions and orthogonal groups
186
8.10 Algebraic topology and orthogonal groups
190
8.11 Canonical volume form, mixed product and cross product . . 193
8.12 Exercises
196
C h a p t e r 9. Euclidean affine spaces
9.1 Definitions. Isometries and rigid motions
9.2 Orthogonal subspaces. Distances between subspaces . . . .
9.3 Structure of isometries. Generators of Is(X) and Is + (X) . .
9.4 Polygonal billiards and the structure of plane isometries . .
9.5 Similarities
9.6 Plane similarities
9.7 Distances between several points
9.8 Stabilizers of subsets
9.9 Length of curves
9.10 Distance and differential geometry. The first variation formula
9.11 The Hausdorff metric
9.12 Canonical measure and volume
9.13 Steiner symmetrization
9.14 Exercises
,
200
202
204
207
212
219
228
236
245
248
252
255
258
264
268
C h a p t e r 10. Triangles, spheres and circles
10.1 Triangles: definitions and notation
10.2 Classical results
10.3 Formulary
10.4 Inequalities and problems of minimum
10.5 Polygons
_
10.6 Tetrahedra
279
280
282
284
288
293
294
viii
Table of contents
10.7
10.8
10.9
10.10
10.11
10.12
10.13
Spheres
Inversion
Circles in the plane
Pencils of circles
Classical problems
Parataxis: prelude to sections 18.9, 20.5 and 20.7
Exercises
296
304
308
313
•• •
317
320
323
C h a p t e r 11. Convex sets
11.1 Definition and examples
11.2 The dimension of a convex set
11.3 Topology of convex sets
11.4 Convex sets and hyperplanes. Separation theorems
11.5 Supporting hyperplanes; applications
11.6 The frontier of a convex set
11.7 Helly's theorem and applications
11.8 Convex functions
11.9 Exercises
331
332
340
341
349
353
360
363
369
381
Bibliography
385
Index of n o t a t i o n s
401
Index
405
Volume II ,
C h a p t e r 12. Polytopes; compact convex sets
12.1-Definitions and examples
12.2 Volume of polytopes
12.3 Area of polytopes
12.4 Regular polygons
12.5 Regular polytopes: definition and examples
12.6 Classification of regular polytopes
12.7 Euler's formula
12.8 Cauchy's theorem
12.9 Approximation of compact convex sets by polytopes . . . .
12.10 Area of compact convex sets
12.11 The isoperimetric inequality
12.12 Exercises .
1
3
13
16
21
24
36
43
50
55
59
70
79
C h a p t e r 13. Quadratic forms
13.1 Definitions and examples
13.2 Isotropy, singularity, radicals and degeneracy
86
88
91
Table of contents
13.3 Orthogonality. The non-singular completion of a subspace . .
13.4 Orthogonalization. Classification of real and complex quadratic
forms
13.5 Simultaneous orthogonalization of two quadratic forms . . .
13.6 The group of a quadratic form. Generalities
13.7 Witt's theorem and the theorem of Cartan-Dieudonne . . .
13.8 Isometries of Artinian planes
13.9 Exercises
ix
94
97
99
101
105
110
114
Chapter 14. Projective quadrics
14.1 Definitions and examples
14.2 Subspaces of PQ{E). Pencils of quadrics
14.3 Topology of real and complex quadrics
14.4 Quadrics in Art4
14.5 Duality with respect to a proper quadric. Polarity
14.6 Tangential quadrics and tangential equation
14.7 The group of a proper quadric
14.8 Exercises
116
117
122
125
129
134
139
142
143
Chapter 15. Affine quadricB
15.1 Definitions and notation
15.2 Reduction of affine quadratic forms
15.3 Classification of real and complex affine quadrics
15.4 The topology of real and complex affine quadrics
15.5 Polarity with respect to a proper affine quadric
15.6 Euclidean affine quadrics
15.7 Exercises
146
147
149
150
157
159
163
166
Chapter 16. Projective conies
170
16.1 Notation and expressions
171
16.2 Good parametrizations, cross-ratios and Pascal's theorem . . 173
16.3 Homographies and group of a conic. Applications
178
16.4 Intersection of two conies. Bezout's theorem
182
16.5 Pencils of conies
192
16.6 The great Poncelet theorem
203
16.7 Affine conies
210
16.8 Exercises
213
Chapter 17. Euclidean conies
17.1 Descartes's principle
17.2 Metrical properties (elementary)
17.3 Metrical properties (spatial)
17.4 Metrical properties (projective)
17.5 Pencils of conies and the cyclic points
17.6 Tangential pencils of conies. Homofocal conies
218
219
221
227
228
232
239
X
Table of contents
17.7 Specific properties of ellipses
17.8 Specific properties of hyperbolas
17.9 Exercises
245
248
250
C h a p t e r 18. T h e sphere for its own sake
18.1 Definitions, special dimensions, charts and projections
18.2 The topology of spheres
18.3 The canonical measure on the sphere
18.4 The intrinsic metric on the sphere
18.5 The isometry group of the sphere
18.6 Spherical triangles
18.7 Convex spherical polygons and Cauchy's lemma
18.8 The three-sphere and Clifford parallelism
18.9 Villarceau circles and parataxy
18.10 The Mobius group . .
18.11 Exercises
255
. . . 257
273
276
279
281
284
292
298
304
305
311
C h a p t e r 19. Elliptic and hyperbolic geometry
19.1 Elliptic geometry
19.2 The projective model and the ball model
19.3 The fundamental formula
19.4 The isometry group
19.5 The canonical measure of hyperbolic space
19.6 The Poincare model
19.7 Other models
19.8 Exercises
318
320
326
329
331
334
336
343
345
C h a p t e r 20. The space of spheres
20.1 The space of generalized spheres
20.2 The fundamental quadratic form
20.3 Orthogonality
20.4 Intersection of two spheres
20.5 Pencils
20.6 The circular group
20.7 Polyspheric coordinates
20.8 Exercises
349
350
351
353
354
356
358
359
361
"
Bibliography
363
Index of n o t a t i o n s
379
Index
381

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