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Bonahon F. Ч Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots (Student Mathematical Library: Ias Park City Mathematical Subseries)
Bonahon F. Ч Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots (Student Mathematical Library: Ias Park City Mathematical Subseries)



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Ќазвание: Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots (Student Mathematical Library: Ias Park City Mathematical Subseries)

јвтор: Bonahon F.

јннотаци€:

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2009

 оличество страниц: 392

ƒобавлена в каталог: 12.09.2014

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$B_{d}(P, r)$, a ball in a metric space      4
$Id_{X}$, the identity map of X      357
$\mathbb{B}^{2}$, the disk model for the hyperbolic plane      36 36Ч39 43 44 101 158Ч160 209 212 216 223
$\mathbb{C}$, the set of all complex numbers      356 361Ч364
$\mathbb{H}^{2}$, the hyperbolic plane      11 11Ч46
$\mathbb{H}^{3}$, the 3-dimensional hyperbolic space      227 227Ч240
$\mathbb{N}$, the set of all positive integers      356
$\mathbb{Q}$, the set of all rational numbers      356
$\mathbb{R}$, the set of all real numbers      356
$\mathbb{R}^{2}$, the euclidean plane      1Ч10 357
$\mathbb{R}^{3}$, the 3-dimensional euclidean space      4 357
$\mathbb{S}^{2}$, the 2-dimensional sphere      47
$\mathbb{S}^{3}$, the 3-dimensional sphere      343 351
$\mathbb{Z}$, the set of all integers      356
Absolute value      23 362
Action of a group      185 185Ч205 248
Adjacent tiles      137 137Ч146 261
Alexander, James Waddell II (1888-1971)      318
Antilinear fractional map      27 27Ч33 38 186 231Ч235
Antipodal points      49 102
Arc length      see "Length"
Area      131
Area, hyperbolic area      43 43Ч44
Area, spherical area      53 182
Ball, ball in $\mathbb{R}^{3}$      5 344
Ball, ball in a metric space      4 42Ч43 68Ч79 240 262
Ball, ball model for the hyperbolic space      236 240
Belong to      355
Bijection      5 90 185 357
Bijective      357
Bonahon, Francis      iii
Bonnet, Pierre Ossian (1819-1892)      131
Boundary, boundary of a manifold      348
Boundary, boundary point      152
Bounded      62 81 152 152 161
Brock, Jeffrey F.      283
Bromberg, Kenneth W.      282
Busemann, Busemann function      178 183
Busemann, Busemann, Herbert (1905-1994)      178
Canary, Richard D.      283
Cannon, James W.      302
Canonical tile      143 143Ч146
Cauchy, Cauchy sequence      135 182
Cauchy, Cauchy, Augustin (1789-1857)      182 211
Cayley, Arthur (1821-1895)      45
Center, center of a horocircle      172 214
Center, center of a horosphere      234
Cerf, Jean      318
circle      5 17Ч22 32Ч33 38 48 85Ч87 172 207Ч212 230 288 317
Circle, great circle      48 129
Circle, great circle arc      49 50Ч51
Closed, closed curve      49 316
Closed, closed geodesic      49 127Ч129 204
Closed, closed subset      62 80 152 152Ч154 250Ч251 257Ч258 286
Compact metric space      150 150 154 182 203 251 348
Companion knot      321
Complement      356
Complement, knot complement      293Ч313 319Ч340 352Ч354
Complete, complete geodesic      21
Complete, complete metric space      135 147Ч155 174 182 183 203 230 260 311 322 325 341 346 348
Complex number      6 11 23 361 361Ч364
Complex number, complex conjugate      6 23 44 362
Complex number, complex exponential      242 364 362 364
composition      6 15 28 42 51 52 185 186 357
Cone      86
Cone, surface with cone singularities      87 125Ч126 136 192 205 342
conjugate      see "Complex conjugate"
Connected      64 341
Continued fraction      224
Continuous function      4 8 9 61 90 151 302 360
Convergence      4 135 147Ч155 360
Convergence, limit at infinity      361 360Ч361
Convergence, toward infinity      361 360Ч361
Convex, euclidean polygon      62
Convex, hyperbolic polygon      82
Convex, spherical polygon      82
Crossratio      44Ч45
Differential map      29 29Ч38 40 49Ч50 229Ч230 317
Dihedral angle      235 259 259Ч265 290
Dihedron      235
Dirichlet, Dirichlet domain      198 197Ч201
Dirichlet, Lejeune Dirichlet, Gustav (1805-1859)      198
Discontinuity domain      286Ч287
Discontinuous group action      189 189Ч197 203Ч205 248 254 260 268 280Ч283 286 313
Discrete walk      58 58Ч60 69Ч79 85Ч86
Disjoint sets      356
Disk, disk in the plane      5
Disk, disk model for the hyperbolic plane      36 36Ч39 43 44 101 158Ч160 209 212 216 223
Disk, disk sector      72 86 205
Distance, distance function      3
Distance, euclidean distance      2 1Ч3 8 340
Distance, hyperbolic distance      12 12Ч14 37 39 44 45 228
Distance, signed distance      218Ч220 253 253Ч254
Distance, spherical distance      48 343
Domino diagram      225 291
Dumas, David      282
Dunne, Edward G.      xv
Earle, Clifford J.      280
Edge of a hyperbolic polyhedron      258 258Ч265
Edge of a polygon      61 61Ч83 257
Edge, edge cycle      172 172Ч181
Element of a set      355
Elliptic isometry      41 239 240
Empty set      355
Epstein, David B.A.      340
Equivalence class      84
Equivalence relation      84
Essential surface      343Ч346
Essential surface, essential Klein bottle      345
Essential surface, essential projective plane      344
Essential surface, essential sphere      344
Essential surface, essential torus      344
Euclidean, euclidean distance      2 3 340
Euclidean, euclidean geodesic      3 21
Euclidean, euclidean isometries      5Ч7 39
Euclidean, euclidean length      2 48
Euclidean, euclidean metric      67 340
Euclidean, euclidean norm      33
Euclidean, euclidean plane      1Ч10
Euclidean, euclidean polygon      61 61Ч66 183
Euclidean, euclidean space      340 349
Euclidean, euclidean surface      67 89Ч97 103Ч104 341 343
Euclidean, euclidean triangle      125
Euler, Euler characteristic      131
Euler, Euler exponential notation      6 362
Euler, Euler, Leonhard (1707-1783)      131 362
Exceptional fiber      347
Exponential, complex exponential      242 364 362Ч364
Exterior point      152
Face of a polyhedron      258 257Ч265 334Ч340
Farey, crooked Farey tessellation      243 241Ч247 265Ч265 279 291Ч297
Farey, Farey circle packing      210 207Ч226
Farey, Farey series      223
Farey, Farey sum      209
Farey, Farey tessellation      210 207Ч226
Farey, Farey, John (1766-1826)      210
Fiber      347
Fibration      see "Seifert fibration"
Fields medal      322 350
Figure-eight knot      303 293Ч313 352 353
Finite volume      324
fix      358
Fixed point      358
Ford, Ford circles      212
Ford, Ford domain      331 326Ч340
Ford, Ford, Lester Randolph (1886-1967)      212 340
Free group action      192 264 280Ч282 297 352
Fuchs, Lazarus Immanuel (1833-1902)      284
Fuchsian group      255 252Ч257 280 283Ч284
Fuchsian group, fuchsian group of the first type      256
Fuchsian group, fuchsian group of the second type      256
Fuchsian group, twisted fuchsian group      288
Function      357
Fundamental domain      192 185Ч205 259 259Ч269 296Ч298
Gauss Ч Bonnet formula      131
Gauss, Johann Carl Friedrich (1777-1855)      131
Generate transformation group generated by bijections      135Ч136 186 194 260
Genus      97 129
Geodesic      21
Geodesic, closed geodesic      49 127Ч129 204
Geodesic, complete geodesic      21
Geodesic, euclidean geodesic      3 21
Geodesic, hyperbolic geodesic      17 22 230
Geodesic, spherical geodesic      49Ч51
Geometric structure      349
Gordon, Cameron McA.      340
Great circle      48 129
Great circle arc      49
Group      185 185Ч205
Group, abstract group      202
Group, fuchsian group      255 252Ч257 280 283Ч284
Group, group action      185 185Ч205
Group, group of isometries      186
Group, isometry group      186
Group, kleinian group      248 241Ч302 326Ч340
Group, tiling group      136 135Ч184 212Ч222 259
Group, transformation group      185 185Ч205
Haken, Wolfgang      350
Hamilton, Richard S.      350
Haros, C.      211
Hatcher, Allen E.      318
Hiatt, Christopher      xvi
Homeomorphic      90
Homeomorphism      41 90 91 96Ч98 126 129 130 182 270 288 303Ч311 317Ч319 325
Homogeneous      7 15Ч16 34Ч36 49Ч50 128 129 230 349
Homothety      15 28 40 228 239 286
Horizontal translation      15 28 41 228 239
Horocircle      172 172Ч181 214Ч222 290
Horocycle      172
Horocyclic isometry      172 172Ч176 221Ч222
Horodisk      178 178Ч180 256
Horosphere      234 234Ч235 260Ч265 268 298 331Ч332
Howe, Roger E.      xv
Hyperbolic, hyperbolic area      43 43Ч44
Hyperbolic, hyperbolic disk      42
Hyperbolic, hyperbolic distance      12 12Ч14 37 39 44 45 228
Hyperbolic, hyperbolic geodesic      17 22 230
Hyperbolic, hyperbolic isometry      14Ч17 25 23Ч27 40 232 231Ч231 234 238 239
Hyperbolic, hyperbolic knot      322 322Ч340
Hyperbolic, hyperbolic length      11 17Ч22 33Ч34 227 230
Hyperbolic, hyperbolic metric      12 12Ч14 37 39 44 45 82 228
Hyperbolic, hyperbolic norm      33 33Ч36 228 229
Hyperbolic, hyperbolic plane      11 11Ч46 234
Hyperbolic, hyperbolic polygon      80
Hyperbolic, hyperbolic reflection      15 240
Hyperbolic, hyperbolic rotation      40
Hyperbolic, hyperbolic space      227 227Ч240 349
Hyperbolic, hyperbolic surface      82 102 341Ч343
Hyperbolic, hyperbolic triangle      43 98 125 130
Hyperbolic, hyperbolic volume      240 353
Ideal, ideal polygon      116
Ideal, ideal vertex      116 170 170Ч180 257 258 257Ч265
identity map      185 357
Image      357
Imaginary part      11 362
Infimum (pl. infima)      359 358Ч360
Infinite limit      361 360Ч361
Infinity, limit at infinity      361 360Ч361
Infinity, vertex at infinity      170 170Ч180 257 258 257Ч265
Injective      357
Interior, interior of a manifold      348
Interior, interior point      152
Intersection      356
Inverse map      185 357
Inversion across a circle      16 16Ч17 28 28 42
Inversion across a sphere      229 237
Isometric, isometric extension      233 231Ч233 236
Isometric, isometric group action      186
Isometric, locally isometric      67
Isometry      5
Isometry, elliptic isometry      41 239
Isometry, euclidean isometries      5 7 39
Isometry, group of isometries      186
Isometry, horocyclic isometry      172 172Ч176 221Ч222
Isometry, hyperbolic isometry      14Ч17 25 23Ч27 40 232 231Ч234 238 239
Isometry, isometry group      186
Isometry, loxodromic isometry      40 239
Isometry, parabolic isometry      41 239 327
Isometry, spherical isometries      50
Isomorphic knots      317
Isotopic knots      316
Isotopy      316
Isotropic      7 34Ч36 49Ч50 230 349
Jaco, William H.      321 350
Jacobian      238 317
Johannson, Klaus      321 350
Klein, Klein bottle      95 93Ч95 128 341
Klein, Klein bottle, essential Klein bottle      345
Klein, Klein, Felix (1849-1925)      45 95 284
Kleinian group      248 241Ч302 326Ч340
Kneser, Hellmuth (1898-1973)      350
Knot      316
Knot, companion knot      321
Knot, figure-eight knot      303
Knot, hyperbolic knot      322 322Ч340
Knot, knot complement      293Ч313 319Ч340 352Ч354
Knot, satellite knot      321
Knot, torus knot      319
Knot, trefoil knot      319
Knot, unknot      317
Length, euclidean length      2 48
Length, hyperbolic length      11 17Ч22 33Ч34 227 230
Length, length in a metric space      3 9 87 127
Length, length of a discrete walk      58
Length, length of a sequence      135 147Ч155
LIMIT      4 135 147Ч155
Limit, limit at infinity      361 360 361
Limit, limit point      248
Limit, limit set      248 247Ч252 255Ч256 270Ч279 299Ч302
Linear fractional map      27 27 33 38 186 231Ч235
Linear groups      202
Listing, Johann Benedict (1808-1882)      104
Little, Charles N.      337
Locally finite      134 145 197 259 261 333 334
Locally isometric      67 82 83
Loxodromic isometry      40 239
Luecke, John      340
Manifold      340 340Ч351
MAP      357
Marden, Albert      296
Maximum (pl. maxima)      358 358Ч360
McMullen, Curtis T.      322
Metric      3
Metric, euclidean metric      67 340
Metric, hyperbolic metric      12 12Ч14 37 39 44 45
Metric, metric function      3
Metric, metric space      3 3Ч5 58Ч61 134Ч135 150Ч155
Metric, path metric      9 63 81 82
Metric, product metric      8 344
Metric, quotient metric      61 58 61 141Ч143 187Ч189
Metric, spherical metric      48 343
Milnor, John W.      350
Minimum (pl. minima)      358 358Ч360
Minsky, Yair N.      282 283
Moduli space      342
Modulus (pl. moduli)      23 362
Moebius strip      341
Moebius, Moebius group      237
Moebius, Moebius strip      104 103Ч104
Moebius, Moebius transformation      237
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