Berger Geometry Revealed

A Jacob's Ladder to Modern Higher Geometry
1. Auflage 2010
ISBN: 978-3-540-70997-8
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 831 Seiten

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built 'above' the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called 'elementary' geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Marcel Berger is Ancien Professeur of the University of Paris and emeritus director of research at the Centre National de la Recherche Scientifique (CNRS), from 1979 to 1981 he was president of the French Mathematical Society and from 1985 to 1994 director of the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette.

Autoren/Hrsg.

Berger, Marcel

Weitere Mitwirkende

Senechal, Lester J.

Inhaltsverzeichnis

1;About the Author;6
2;Introduction;8
2.1;Bibliography;11
3;Table of Contents;12
4;I Points and lines in the plane;18
4.1;I.1 In which setting and in which plane are we working? And right away an utterly simple problem of Sylvester about the collinearity of points;18
4.2;I.2 Another naive problem of Sylvester, this time on the geometric probabilities of four points;23
4.3;I.3 The essence of affine geometry and the fundamental theorem;29
4.4;I.4 Three configurations of the affine plane and what has happened to them: Pappus, Desargues and Perles;34
4.5;I.5 The irresistible necessity of projective geometry and the construction of the projective plane;40
4.6;I.6 Intermezzo: the projective line and the cross ratio;45
4.7;I.7 Return to the projective plane: continuation and conclusion;48
4.8;I.8 The complex case and, better still, Sylvester in the complex case: Serre's conjecture;57
4.9;I.9 Three configurations of space (of three dimensions): Reye, Möbius and Schläfli;60
4.10;I.10 Arrangements of hyperplanes;64
4.11;I. XYZ;65
4.12;Bibliography;74
5;II Circles and spheres;77
5.1;II.1 Introduction and Borsuk's conjecture;77
5.2;II.2 A choice of circle configurations and a critical view of them;82
5.3;II.3 A solitary inversion and what can be done with it;94
5.4;II.4 How do we compose inversions? First solution: the conformal group on the disk and the geometry of the hyperbolic plane;98
5.5;II.5 Second solution: the conformal group of the sphere, first seen algebraically, then geometrically, with inversions in dimension 3(and three-dimensional hyperbolic geometry). Historical appearanceof the first fractals;103
5.6;II.6 Inversion in space: the sextuple and its generalization thanksto the sphere of dimension 3;107
5.7;II.7 Higher up the ladder: the global geometry of circles and spheres;112
5.8;II.8 Hexagonal packings of circles and conformal representation;119
5.9;II.9 Circles of Apollonius;129
5.10;II. XYZ ;132
5.11;Bibliography;153
6;III The sphere by itself: can we distribute points on it evenly?;156
6.1;III.1 The metric of the sphere and spherical trigonometry;156
6.2;III.2 The Möbius group: applications;162
6.3;III.3 Mission impossible: to uniformly distribute points on the sphere S2: ozone, electrons, enemy dictators, golf balls, virology, physics of condensed matter;164
6.4;III.4 The kissing number of S2, alias the hard problem of the thirteenth sphere;185
6.5;III.5 Four open problems for the sphere S3;187
6.6;III.6 A problem of Banach--Ruziewicz: the uniqueness of canonical measure;189
6.7;III.7 A conceptual approach for the kissing number in arbitrary dimension;190
6.8;III. XYZ ;192
6.9;Bibliography;193
7;IV Conics and quadrics;196
7.1;IV.1 Motivations, a definition parachuted from the ladder, and why;196
7.2;IV.2 Before Descartes: the real Euclidean conics. Definition and some classical properties;198
7.3;IV.3 The coming of Descartes and the birth of algebraic geometry;213
7.4;IV.4 Real projective theory of conics; duality;215
7.5;IV.5 Klein's philosophy comes quite naturally;220
7.6;IV.6 Playing with two conics, necessitating once again complexification;223
7.7;IV.7 Complex projective conics and the space of all conics;227
7.8;IV.8 The most beautiful theorem on conics: the Poncelet polygons;231
7.9;IV.9 The most difficult theorem on the conics: the 3264 conics of Chasles;241
7.10;IV.10 The quadrics;247
7.11;IV. XYZ;257
7.12;Bibliography;260
8;V Plane curves;263
8.1;V.1 Plain curves and the person in the street: the Jordan curve theorem, the turning tangent theorem and the isoperimetric inequality;263
8.2;V.2 What is a curve? Geometric curves and kinematic curves;268
8.3;V.3 The classification of geometric curves and the degree of mappings of the circle onto itself;271
8.4;V.4 The Jordan theorem;273
8.5;V.5 The turning tangent theorem and global convexity;274
8.6;V.6 Euclidean invariants: length (theorem of the peripheral boulevard) and curvature (scalar and algebraic): Winding number;277
8.7;V.7 The algebraic curvature is a characteristic invariant: manufacture of rulers, control by the curvature;283
8.8;V.8 The four vertex theorem and its converse; an application to physics;285
8.9;V.9 Generalizations of the four vertex theorem: Arnold I;292
8.10;V.10 Toward a classification of closed curves: Whitney and Arnold II;295
8.11;V.11 Isoperimetric inequality: Steiner's attempts;309
8.12;V.12 The isoperimetric inequality: proofs on all rungs;312
8.13;V.13 Plane algebraic curves: generalities;319
8.14;V.14 The cubics, their addition law and abstract elliptic curves;322
8.15;V.15 Real and Euclidean algebraic curves;334
8.16;V.16 Finite order geometry;342
8.17;V. XYZ ;345
8.18;Bibliography;350
9;VI Smooth surfaces;355
9.1;VI.1 Which objects are involved and why? Classification of compact surfaces;355
9.2;VI.2 The intrinsic metric and the problem of the shortest path;359
9.3;VI.3 The geodesics, the cut locus and the recalcitrant ellipsoids;361
9.4;VI.4 An indispensable abstract concept: Riemannian surfaces;371
9.5;VI.5 Problems of isometries: abstract surfaces versus surfaces of E3;375
9.6;VI.6 Local shape of surfaces: the second fundamental form, total curvature and mean curvature, their geometric interpretation, the theorema egregium, the manufacture of precise balls;378
9.7;VI.7 What is known about the total curvature (of Gauss);387
9.8;VI.8 What we know how to do with the mean curvature, all about soap bubbles and lead balls;394
9.9;VI.9 What we don't entirely know how to do for surfaces;400
9.10;VI.10 Surfaces and genericity;405
9.11;VI.11 The isoperimetric inequality for surfaces;411
9.12;VI. XYZ ;413
9.13;Bibliography;417
10;VII Convexity and convex sets;422
10.1;VII.1 History and introduction;422
10.2;VII.2 Convex functions, examples and first applications;425
10.3;VII.3 Convex functions of several variables, an important example;428
10.4;VII.4 Examples of convex sets;430
10.5;VII.5 Three essential operations on convex sets;433
10.5.1;VII.5.A The (Steiner, Schwarz) symmetrizations;433
10.5.2;VII.5.B Some algebra of convex sets: Minkowski's sum;437
10.5.3;VII.5.C A duality: polarity;438
10.6;VII.6 Volume and area of (compacts) convex sets, classical volumes: Can the volume be calculated in polynomial time?;441
10.6.1;VII.6.A Volume of cubes, cocubes and simplexes;442
10.6.2;VII.6.B Balls, spheres and ellipsoids;443
10.6.3;VII.6.C Approximation by polytopes, areas of convex sets;447
10.6.4;VII.6.D Mission impossible: calculating the volume of a convex set numerically;448
10.7;VII.7 Volume, area, diameter and symmetrizations: first proof of the isoperimetric inequality and other applications;450
10.8;VII.8 Volume and Minkowski addition: the Brunn-Minkowski theorem and a second proof of the isoperimetric inequality;452
10.9;VII.9 Volume and polarity;457
10.10;VII.10 The appearance of convex sets, their degree of badness;459
10.10.1;VII.10.A How to generate a convex set;459
10.10.2;VII.10.B Topology of complex sets and their boundaries;461
10.10.3;VII.10.C The John-Loewner ellipsoid and its applications;461
10.10.4;VII.10.D A first metric space formed of all center-symmetric convex sets: the compact set of Banach-Mazur;464
10.10.5;VII.10.E The Rogalski conjecture and a mapping of isoperimetric type;467
10.10.6;VII.10.F Badness test for a convex set using its moments of inertia: the ellipsoids of Legendre and Binet (ellipsoid of inertia), the inertial invariant and the grand conjecture on convex sets (first formulation);469
10.11;VII.11 Volumes of slices of convex sets;472
10.11.1;VII.11.A Slicing by lines, Hammer's X-ray problem;473
10.11.2;VII.11.B Slicing with hyperplanes: general questions, the grand conjecture;474
10.11.3;VII.11.C The hyperplane sections of the cube;478
10.12;VII.12 Sections of low dimension: the concentration phenomenonand the Dvoretsky theorem on the existence of almostspherical sections;483
10.12.1;VII.12.A Statement of the result;483
10.12.2;VII.12.B The concentration phenomenon of Paul Lévy;486
10.12.3;VII.12.C The proof;488
10.13;VII.13 Miscellany;490
10.13.1;VII.13.A Projections;490
10.13.2;VII.13.B Steiner-Minkowski formula and mixed volume;491
10.13.3;VII.13.C Convex sets and mathematical physics: the floating body that loses its head, the fundamental frequency, the Poinsot motion, Newtonian gravitation, the destiny of the rolling stones;492
10.13.4;VII.13.D The appearance of the boundary of a convex set; the space of all convex sets;502
10.13.5;VII.13.E Immobilization of a convex set;505
10.14;VII.14 Intermezzo: can we dispose of the isoperimetric inequality?;506
10.15;Bibliography;512
11;VIII Polygons, polyhedra, polytopes;518
11.1;VIII.1 Introduction;518
11.2;VIII.2 Basic notions;519
11.3;VIII.3 Polygons;521
11.4;VIII.4 Polyhedra: combinatorics;526
11.5;VIII.5 Regular Euclidean polyhedra;531
11.6;VIII.6 Euclidean polyhedra: Cauchy rigidity and Alexandrov existence;537
11.7;VIII.7 Isoperimetry for Euclidean polyhedra;543
11.8;VIII.8 Inscribability properties of Euclidean polyhedra; how to encage a sphere (an egg) and the connection with packings of circles;545
11.9;VIII.9 Polyhedra: rationality;550
11.10;VIII.10 Polytopes (d4): combinatorics I;552
11.11;VIII.11 Regular polytopes (d4);557
11.12;VIII.12 Polytopes (d4): rationality, combinatorics II;563
11.13;VIII.13 Brief allusions to subjects not really touched on;568
11.14;Bibliography;571
12;IX Lattices, packings and tilings in the plane;575
12.1;IX.1 Lattices, a line in the standard lattice Z2 and the theory of continued fractions, an immensity of applications;575
12.2;IX.2 Three ways of counting the points Z2 in various domains: pick and Ehrhart formulas, circle problem;579
12.3;IX.3 Points of Z2 and of other lattices in certain convex sets: Minkowski's theorem and geometric number theory;585
12.4;IX.4 Lattices in the Euclidean plane: classification, density, Fourier analysis on lattices, spectra and duality;588
12.5;IX.5 Packing circles (disks) of the same radius, finite or infinite in number, in the plane (notion of density). Other criteria;598
12.6;IX.6 Packing of squares, (flat) storage boxes, the grid (or beehive) problem;605
12.7;IX.7 Tiling the plane with a group (crystallography). Valences, earthquakes;608
12.8;IX.8 Tilings in higher dimensions;615
12.9;IX.9 Algorithmics and plane tilings: aperiodic tilings and decidability, classification of Penrose tilings;619
12.10;IX.10 Hyperbolic tilings and Riemann surfaces;629
12.11;Bibliography;632
13;X Lattices and packings in higher dimensions;635
13.1;X.1 Lattices and packings associated with dimension 3;635
13.2;X.2 Optimal packing of balls in dimension 3, Kepler's conjecture at last resolved;641
13.3;X.3 A bit of risky epistemology: the four color problem and the Kepler conjecture;651
13.4;X.4 Lattices in arbitrary dimension: examples;653
13.5;X.5 Lattices in arbitrary dimension: density, laminations;660
13.6;X.6 Packings in arbitrary dimension: various options for optimality;666
13.7;X.7 Error correcting codes;671
13.8;X.8 Duality, theta functions, spectra and isospectrality in lattices;679
13.9;Bibliography;685
14;XI Geometry and dynamics I: billiards;687
14.1;XI.1 Introduction and motivation: description of the motion of two particles of equal mass on the interior of an interval;687
14.2;XI.2 Playing billiards in a square;691
14.2.1;XI.2.A The dichotomy and continued fractions;692
14.2.2;XI.2.B Counting periodic trajectories;697
14.2.3;XI.2.C Introduction of the language of dynamical systems;700
14.3;XI.3 Particles with different masses: rational and irrational polygons;701
14.4;XI.4 Results in the case of rational polygons: first rung;704
14.5;XI.5 Results in the rational case: several rungs higher on the ladder;708
14.5.1;XI.5.A The nature of nonperiodic trajectories;709
14.5.2;XI.5.B Counting periodic trajectories;715
14.6;XI.6 Results in the case of irrational polygons;717
14.7;XI.7 Return to the case of two masses: summary;722
14.8;XI.8 Concave billiards, hyperbolic billiards;722
14.9;XI.9 Circles and ellipses;725
14.10;XI.10 General convex billiards;729
14.10.1;XI.10.A Very smooth and strictly convex billiards: caustics;729
14.10.2;XI.10.B Three strange phenomena;731
14.10.3;XI.10.C Generic billiards;735
14.10.4;XI.10.D Periodic trajectories;737
14.10.5;XI.10.E Billiards and duality;739
14.11;XI.11 Billiards in higher dimensions;740
14.12;XI.XYZ Concepts and language of dynamical systems;742
14.12.1;XI.XYZ.A Ergodicity and mixing;742
14.12.2;XI.XYZ.B The various notions of entropy;745
14.13;Bibliography;747
15;XII Geometry and dynamics II: geodesic flow on a surface;750
15.1;XII.1 Introduction;750
15.2;XII.2 Geodesic flow on a surface: problems;752
15.3;XII.3 Some examples for sensing the difficulty of the problem;754
15.3.1;XII.3.A The spheres;754
15.3.2;XII.3.B The surfaces of revolution: the Zoll surfaces;754
15.3.3;XII.3.C Weinstein's counterexample;757
15.3.4;XII.3.D Ellipsoids with three axes;758
15.3.5;XII.3.E The flat tori;761
15.4;XII.4 Existence of a periodic trajectory;762
15.4.1;XII.4.A The torus and surfaces of higher genus;762
15.4.2;XII.4.B The sphere, Birkhoff's result;763
15.5;XII.5 Existence of more than one, of many periodic trajectories;and can we count them?;768
15.5.1;XII.5.A The case of the torus;769
15.5.2;XII.5.B Surfaces of higher genus;771
15.5.3;XII.5.C The sphere: the three Lusternik-Schnirelman geodesics.;775
15.5.4;XII.5.D The sphere: an infinity of periodic geodesics;778
15.6;XII.6 What behavior can be expected for other trajectories?Ergodicity, entropies;783
15.6.1;XII.6.A Surfaces of higher genus;783
15.6.2;XII.6.B The entropies;785
15.6.3;XII.6.C The case of the sphere. The example of Osserman-Donnay;786
15.6.4;XII.6.D Entropy and the length of geodesics joining two given points;789
15.7;XII.7 Do the mechanics determine the metric?;790
15.8;XII.8 Recapitulation and open questions;792
15.9;XII.9 Higher dimensions;792
15.10;Bibliography;793
16;Selected Abbreviations for Journal Titles;795
17;Name Index;798
18;Subject Index;804
19;Symbol Index;836