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Ratcliffe J.G. — Foundations of Hyperbolic Manifolds
Ratcliffe J.G. — Foundations of Hyperbolic Manifolds



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Название: Foundations of Hyperbolic Manifolds

Автор: Ratcliffe J.G.

Аннотация:

This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds.

The second edition contains hundreds of changes and corrections, and new additions include: A more thorough discussion of polytopes; Discussion of Simplex Reflection groups has been expanded to give a complete classification of the Gram matrices of spherical, Euclidean and hyperbolic n-simplices; A new section on the volume of a simplex, in which a derivation of Schlafli's differential formula is presented; A new section with a proof of the n-dimensional Gauss-Bonnet theorem.

The exercises have been thoroughly reworked, pruned, and upgraded, and over 100 new exercises have been added. The author has also prepared a solutions manual which is available to professors who choose to adopt this text for their course.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2006

Количество страниц: 796

Добавлена в каталог: 10.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$B^n$      119 123
$C^1$ curve      30
$C^1$ topology      564 599
$D^n$      188
$E^n$      15
$GL(n, \mathbb{C})$      145
$GL(n, \mathbb{R})$      146
$H^n$      61 63
$LF(\hat{\mathbb{C}})$      115
$M(B^n)$      120
$M(S^n)$      119
$M(U^n)$      116
$M(\hat{E}^n)$      111
$PSL(2, \mathbb{C})$      148
$PSL(2, \mathbb{R})$      148
$P^n$      41
$SL(n, \mathbb{C})$      146
$SL(n, \mathbb{R})$      146
$S^n$      35 37
$U^n$      116 131
$\bar{\mathbb{R}}^n$      42
$\eta(x, y)$      59 68 69 71
$\hat{E}^n$      108
$\hat{\mathbb{C}}$      108
$\mathbb{C}^*$      145
$\mathbb{C}^n$      144
$\mathbb{R}^n$      13
$\mathbb{R}^{1,n}$      54
$\mathbb{R}^{n,1}$      54
$\theta (x, y)$      14
(X, G)-atlas of a manifold      347
(X, G)-atlas of an orbifold      691
(X, G)-equivalence of manifolds      369
(X, G)-equivalence of orbifolds      722
(X, G)-manifold      348
(X, G)-manifold structure      348
(X, G)-map of manifolds      356
(X, G)-map of orbifolds      721
(X, G)-orbifold      692
(X, G)-orbifold structure      692
(X, G)-path      701
Abstract polygon      388
Abstract polyhedron      508
Acting, discontinuously      161
Acting, effectively      172
Acting, freely      337
Acting, trivially      172
Action of a group      18
adjacent sides      212
Affine basis      27
Affine combination      27
Affine function      27
Affine hull      27
Affine set      27
Affinely equivalent      344
Affinely independent points      27
Angle of $D^n$      194
Angle of $E^n$      14
Angle of $H^n$      68
Angle of a triangle of $H^2$      80 84
Angle of a triangle of $S^2$      47
Angle sum of a cycle      379
Angles of a hyperbolic triangle      80
Angles of a spherical triangle      47
Angles of rotation      166
Antipodal transformation      16
Antipodal vectors      36
Arc length element of $B^n$      128
Arc length element of $D^n$      192
Arc length element of $E^n$      31
Arc length element of $H^n$      76
Arc length element of $S^n$      43
Arc length element of $U^n$      133
Arc length in a metric space      28
Area of a cusp      427
Area of a disk of $H^2$      79
Area of a disk of $S^2$      46
Area of a triangle of $H^2$      86
Area of a triangle of $S^2$      50
Asymptotic lines      9
Atlas of a manifold      347
Atlas of an orbifold      691
Attractive fixed point      141
Automorphic function      186
Average of a simplex      577
Axis      140 180
B(a, r)      15
Barycentric subdivision      281
Beltrami’s model      7
Bieberbach group      319
Bieberbach theorem      315
Bilinear form      284
Borel measure      565
Borromean rings      459
Boundary of a convex set      195
Boundary of a manifold      335
Boundary of a surface      377
Bounded parabolic limit point      621
C(a, r)      15
Carrier face      209
Cauchy inequality      13
Cauchy sequence      361
Center of a Dirichlet domain      243
Centroid      225
Change of scale      20
Change of scale, equivalent      20
Chart of a manifold      347 348
Chart of an orbifold      691 693
Chordal metric      108
Circle at infinity      7
Circumference of a cusp      427
Circumference of a disk of $H^2$      79
Circumference of a disk of $S^2$      46
Classical projective space      42
Classical Schottky group      613
Clifford parallels      42
Clifford — Klein space-form      341
Closed ball      15
Closed geodesic      404
Closed manifold      334
Closed surface      375
Closure of a convex set      195
Coeffcient vector      113
Collinear points of $E^n$      25
Collinear points of $H^n$      65
Collinear points of $S^n$      38
Combinatorially equivalent      231
Common orthogonal      68
Compact-open topology      150 185
Complete manifold      367
Complete manifold structure      368
Complete metric space      361
Complete orbifold      719
Complete orbifold structure      719
Complex space      144
Conformal ball model      123
Conformal disk model      8
Conformal function      104
Congruence transformation      19
Congruent sets      19
Conical limit point      617
Connected set of polyhedra      254
Connected sum of surfaces      376
Constant (X, G)-path      701 706
Continuation      355
Convex combination      27
Convex core      637
Convex hull      27 210 221 601
Convex polyhedron      201
Convex set      24 194
Coordinate change of a manifold      347
Coordinate change of an orbifold      691
Coordinate frame of $E^n$      26
Coordinate frame of $H^n$      73
Coordinate frame of $S^n$      40
Coordinate neighborhood of a manifold      347
Coordinate neighborhood of an orbifold      691
Coplanar      21
Coset space      147
Coxeter graph      274
Coxeter group      274
Cross product      35
Cross ratio      110
Crystallographic group      310
Curvature of a manifold      371
Curvature of a surface      5
Curve      22
Curve of type $C^1$      30
Cusp      636
Cusp of a manifold      446 673
Cusp of a polyhedron      624
Cusp of a surface      427
Cusp point of a manifold      444 512
Cusp point of a polyhedron      512 624
Cusp point of an orbifold      736
Cusped limit point      621
Cusped region      619 620
Cusped side      631
Cycle of a facet-pairing      509
Cycle of a side-pairing of a polyhedron      255
Cycle of a side-pairing of polygons      379
Cycle of a side-pairing of polyhedra      436 724
Cycle of cusp points      512 736
Cycle of edges      482
Cycle of ideal points      512 736
Cycle of ideal vertices of polygons      421
Cycle of ideal vertices of polyhedra      444
Cycle of polyhedra      256
Cycle of sides      259 421
Cycle relation      259
Cycle transformation      259 423
Cyclic ridge cycle      725
Davis 120-cell space      511
De Rham cohomology      569
Deformation space      396
Dehn surgery      498
Dehn surgery, invariant      497
Dehn — Nielsen theorem      395
Developing map of a manifold      358
Developing map of an orbifold      712
Dihedral angle      213
Dihedral angle sum      436 726
Dihedral ridge cycle      725
Dilogarithm function      466
Dimension of a convex set      195
Dirichlet domain      243
Dirichlet polyhedron      249
Discontinuous action      161
Discontinuous group      162
Discrete faithful representation      396
Discrete group      157
Discrete reflection group      265
Disjoint set of polyhedra      509 724
Distance between sets      234
Dual regular polytope      231
EDGE      234
Edge cycle      436
Edge invariants      482
Edge of an abstract polygon      388
Effective action      172
Elementary group      176 184
Elementary space-form      344
Elliptic closed curve      406
Elliptic elementary group      176
Elliptic geometry      41
Elliptic isometry of $E^n$      165
Elliptic line      41
Elliptic modular group      165 249
Elliptic plane      42
Elliptic space      41
Elliptic transformation of $B^n$      136
Elliptic transformation of $U^n$      137
Elliptic type      176
Erlanger Program      21
Euclidean angle      14
Euclidean distance      14
Euclidean geometry      1 13
Euclidean inner product      13
Euclidean isometry      15
Euclidean line      21
Euclidean manifold      348
Euclidean manifold structure      348
Euclidean metric      14
Euclidean norm      13
Euclidean orbifold      693
Euclidean plane      19
Euclidean similarity      20
Euclidean similarity manifold      348
Euclidean similarity orbifold      693
Euclidean similarity structure      348
Euclidean space      15
Euclidean space-form      341
Euclidean sphere      102 125
Euclidean topology      15
Euclidean torus      343
Euler characteristic of a surface      376
Exact fundamental polyhedron      251
Exact tessellation      253
Extended hyperplane      109
Exterior of light cone      55
Face      206
Facet      508
Facet-pairing      509
Field of fractions      322
Figure-eight knot      450
Figure-eight knot, complement      448
Finitely compact metric space      151
Finitely generated ring      330
First kind      607
Flag      228 233
Free action      337
Freely homotopic      406
Fuchsian group      186
Fundamental cycle      577
Fundamental domain      236
Fundamental orbifold group      704
Fundamental polyhedron      249
Fundamental region      236
Fundamental set      242
Gauss — Bonnet theorem      390 528
General linear group      145
Generalized polytope      226
Generalized simplex      226
Generalized triangle      84
Generalized vertex      221
Genus of a surface      377
Geodesic      26
Geodesic arc      22
Geodesic curve      25
Geodesic half-line      25
Geodesic line      25
Geodesic ray      25
Geodesic section      25
Geodesic segment      23
Geodesically complete      26
Geodesically connected      24
Geodesically convex      24
Geometric manifold      372
Geometric orbifold      692
Geometric space      335
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