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Joyce D.D. Ч Riemannian holonomy groups and calibrated Geometry
Joyce D.D. Ч Riemannian holonomy groups and calibrated Geometry

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Ќазвание: Riemannian holonomy groups and calibrated Geometry

јвтор: Joyce D.D.

јннотаци€:

This graduate level text covers an exciting and active area of research at the crossroads of several different fields in mathematics and physics. In mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in physics String Theory and Mirror Symmetry.
Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kahler structures suitable for beginning graduate students, the text covers seminal results such as Yau's proof of the Calabi Conjecture, and takes the reader all the way to the frontiers of current research in calibrated geometry, giving many open problems.


язык: en

–убрика: ћатематика/

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

ed2k: ed2k stats

√од издани€: 2007

 оличество страниц: 320

ƒобавлена в каталог: 22.05.2008

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$C^{k, \alpha}$      see УHolder spaceФ
$C^{k}$ space      5
$G_2$ holonomy      54Ч55 227Ч239
$G_2$ holonomy, associative 3-fold      68 71 72 254Ч272
$G_2$ holonomy, coassociative 4-fold      68 71 72 254Ч272
$G_2$ holonomy, compact manifolds, Betti numbers of      238
$G_2$ holonomy, compact manifolds, constructing      233Ч239
$G_2$ holonomy, compact manifolds, moduli space of      232 253
$G_2$ holonomy, compact manifolds, topology of      230Ч232
$G_2$ holonomy, explicit metrics      253
$G_2$ holonomy, holonomy subgroups      230
$G_2$ holonomy, metrics Ricci-flat      229
$G_2$ holonomy, parallel spinors      230
$G_2$ instanton      253
$G_2$, definition      227
$G_2$-manifold      228
$G_2$-manifold, with holonomy SU(2) or SU(3)      230
$G_2$-structure      228 252
$G_2$-structure, function $\Theta$      228
$G_2$-structure, nearly parallel      253
$G_2$-structure, positive 3-form      228 253
$G_2$-structure, positive 4-form      228
$G_2$-structure, small torsion      233 235
$G_2$-structure, splitting of forms      229 231
$G_2$-structure, torsion      228
$G_2$-structure, torsion-free      228 233 236
$L^p(M)$      see УLebesgue spaceФ
$L^p_k(M)$      see УSobolev spaceФ
$\mathbb{H}$      see УQuaternionsФ
$\mathbb{O}$      see УOctonionsФ
*      see УHodge starФ
3-Sasakian manifold      222Ч223
3-Sasakian quotient construction      225
A-genus      64 216 242
A-model      184
Affine structure      168 194
Affine structure, integral      195 197 198
Age grading      131Ч133
ALE manifold, Calabi Ч Yau      234 250Ч251
ALE manifold, Eguchi Ч Hanson space      205Ч206 212 234 246
ALE manifold, hyperkahler      201 205Ч207
ALE manifold, Spin(7)      250Ч251
Almost Calabi Ч Yau m-folds      165Ч166
Almost Calabi Ч Yau m-folds, generic      166 173 177 199Ч200
Ambrose Ч Singer Holonomy Theorem      31 36
Associative 3-folds      68 71 72 254Ч272
Associative 3-folds, deformations      256
Associative 3-folds, examples      259Ч264 268Ч270
Associative 3-folds, ruled      261Ч262
Associative cones      258 260 263
Associative cones, and pseudoholomorphic curves in $S^6$      258 261 262
Associative cones, as an integrable system      261
Associative cones, two-sided      258
Associative fibrations      272
Asymptotic cone      152
Asymptotically conical SL m-folds      152 161Ч165
Asymptotically conical SL m-folds, deformations      162
Asymptotically conical SL m-folds, examples      155Ч157 163Ч165
Asymptotically cylindrical manifold      237
Asymptotically Locally Euclidean      see УALE manifoldФ
Atiyah Ч Singer index theorem      18 64 242 275
B-model      184
Berger's theorem      52
Betti number      2 238 248 252
Betti number, refined      59 231 243
Bianchi identities      34 41 43 56
Blow-up      93Ч94
Bochner theorem      60 125
Bootstrap method      117
Brane      180 183 184
Calabi Conjecture      54 85 100Ч121 211 234 237 239
Calabi Ч Yau manifold      54 68 85 100 122Ч145 239
Calabi Ч Yau manifold, A-model      184
Calabi Ч Yau manifold, almost      165Ч166 173 177 199Ч200
Calabi Ч Yau manifold, B-model      184
Calabi Ч Yau manifold, constructions      139Ч144
Calabi Ч Yau manifold, definition      126
Calabi Ч Yau manifold, deformations      144Ч145
Calabi Ч Yau manifold, Hodge numbers      126
Calabi Ч Yau manifold, mirror pair      143 180 183
Calabi Ч Yau orbifold      136
Calibrated geometry      65Ч74 146Ч177 254Ч277
Calibrated submanifold      67
Calibration      67 126
Calibration, classification on $\mathbb{R}^n$      69Ч72
Canonical bundle      96
Cartan Ч Kahler theory      147 153
Category      184Ч185
Category, abelian      185Ч186
Category, additive      185
Category, Ax-      187
Category, Calabi Ч Yau $A_{\infty ^-}$      190
Category, definition      184
Category, derived      186Ч187
Category, derived Fukaya      184 189
Category, dg-      187
Category, enhanced triangulated      190
Category, equivalence      184 185
Category, exact      185
Category, Fukaya      184 187Ч189
Category, functor      184
Category, linear      185
Category, ofcoherent sheaves      184 185 187
Category, triangulated      186Ч187
Category, triangulated $A_{\infty ^-}$      187 189Ч190
Cayley 4-folds      68 71 272Ч277
Cayley 4-folds and associative 3-folds      274
Cayley 4-folds and coassociative 4-folds      274
Cayley 4-folds and holomorphic surfaces      273
Cayley 4-folds and SL 4-folds      273
Cayley 4-folds and Spin(7) instantons      277
Cayley 4-folds, 2-ruled      274
Cayley 4-folds, deformations      273
Cayley 4-folds, examples      273Ч276
Cayley 4-folds, with isolated conical singularities      276
Cayley fibrations      276Ч277
Cayley integral current      277
Cayley numbers      see УOctonionsФ
Characteristic class      96
Characteristic class, A-genus      64 216 242
Characteristic class, first Chern class      96 99 100 123
Characteristic class, first Pontryagin class      232 242 244
Cheeger Ч Gromoll Theorem      61
Chow's theorem      92 97 126
Closed string mirror symmetry      181Ч183 190Ч191
Closed strings      180
Coassociative 4-folds      68 71 72 254Ч272
Coassociative 4-folds, 2-ruled      263Ч264
Coassociative 4-folds, asymptotically conical      263
Coassociative 4-folds, deformations      257
Coassociative 4-folds, examples      259Ч264 268Ч270
Coassociative 4-folds, with isolated conical singularities      270Ч271
Coassociative cones      258 261
Coassociative cones, 2-ruled      263
Coassociative fibrations      271Ч272
Coherent sheaf      184 185 187 190
Cohomology, de Rham      2
Cohomology, Dolbeault      81
Cohomology, Hodge numbers      87
Cohomology, ofsheaves      91 95
Cohomology, other cohomology theories      2
Cohomology, Poincare duality      2 131 132 134 168 232
Compact linear map      6 104
Complete intersection      140
Complex manifold      76Ч79
Complex manifold, biholomorphism      78
Complex manifold, holomorphic function      76
Complex manifold, holomorphic map      78
Complex manifold, rigid      94 133 140
Complex projective space      77
Complex projective space, weighted      134 135 141Ч142 181 190 249
Complex structure      76Ч77
Complex structure, almost      76
Complex symplectic manifold      214Ч216
Complex symplectic manifold, irreducible      214
Complex symplectic manifold, marked      215
Complex symplectic manifold, moduli space      215
Connection      19Ч24
Connection, Levi-Civita      39Ч41
Connection, on principal bundle      22
Connection, on tangent bundle      32Ч36
Connection, on vector bundle      21
Connection, torsion-free      34Ч36
Continuity method      106
Crepant resolution      127Ч133
Current      72
Current, calibrated      73
Current, Cayley      277
Current, integral      72
Current, rectifiable      72
Current, special Lagrangian      158 176
Curvature, and holonomy groups      30Ч32
Curvature, in principal bundles      23
Curvature, in vector bundles      21Ч22
Curvature, of Kahler metrics      84Ч85
Curvature, Ricci      42
Curvature, Riemann      40Ч43 45 50 52 56 58 84 112 125 229 232 235 236 244 247
Curvature, scalar      42
de Rham theorem      2
Deformation      94Ч95
Deformation, of $\mathbb{C}^{m}/G$      133
Deformation, of Calabi Ч Yau manifold      144Ч145
Deformation, smoothing      94
Deformation, universal      95
Deformation, versal      95
Derived category      186Ч187
Derived category, of coherent sheaves      184 187
Derived Fukaya category      169 184
Dg-category      187
Dirac operator      62Ч64 242 243
Discriminant      191
Discriminant, trivalent graph      196
Divisor      98Ч99
Divisor, exceptional      94 128
Divisor, prime      98
Dolbeault cohomology      81
Double point      128
Double point, ordinary      129
Double point, rational      130
Eguchi Ч Hanson space      205Ч206 212 234 246
Elliptic operator      7Ч18
Elliptic operator, $L^p$ estimate      13
Elliptic operator, definition      9 11
Elliptic operator, existence of solutions      16Ч18
Elliptic operator, kernel finite-dimensional      16
Elliptic operator, nonlinear      9 102
Elliptic operator, regularity      12Ч15 66 74 116 121 162 167 171 237 258 266
Elliptic operator, Schauder estimate      13Ч14
Elliptic operator, symbol      9 11
Embedding      65
Enhanced triangulated category      190
Equivalence of categories      185
Exterior differential systems      54 147 153 252 256 264 270 273
Exterior form      see УFormФ
Fano variety      180 190 221
Floer homology, Lagrangian      189
Floer homology, obstructions      169 188Ч189
FLOP      129
FORM      1Ч4
Form, complex symplectic      203
Form, G-structure splitting      56Ч58
Form, Hermitian      82
Form, holomorphic volume      123 126
Form, hyperkahler 2-form      203
Form, Kahler      82 123
Form, of type (p, q)      80
Form, on Kahler manifolds      85Ч86
Frobenius theorem      46
Fukaya category      169 184 187Ч189
Fukaya category, derived      189
functor      184 190
Functor, (left or right) derived      186
Functor, (left or right) exact      186
G-structure      36
Geometric measure theory      72Ч74 158 170 171 176 277
Green's representation      14
Gromov Ч Hausdorff limit      197
Gromov Ч Witten invariants      182Ч183 191
Hilbert scheme      218
HMS Conjecture      184 187Ч190
Hodge numbers      87
Hodge numbers, of hyperkahler manifolds      216
Hodge star      3 57
Hodge star, on Kahler manifolds      86
Hodge theory      3Ч4 58Ч60
Hodge theory, on Kahler manifolds      86Ч88
Holder space      5Ч6
Holder's inequality      4
Holomorphic vector bundle      81 185
Holomorphic volume form      126
Holonomy algebra      27 29 43
Holonomy group and cohomology      56Ч61
Holonomy group and curvature      30Ч32
Holonomy group, classification      52Ч56
Holonomy group, constant tensors      32Ч34
Holonomy group, definition      25 28 42
Holonomy group, exceptional      see У$G_2$ holonomyФ Holonomy УSpin(7) etc.
Holonomy group, for principal bundles      28Ч30
Holonomy group, for vector bundles      24Ч28
Holonomy group, restricted      26 28 42
Holonomy group, Ricci-flat      55
Holonomy group, Riemannian      42Ч43
Homological mirror symmetry      183Ч191
Homological mirror symmetry conjecture      see УHMS ConjectureФ
Hypercomplex algebraic geometry      224
Hypercomplex manifold      223Ч224
Hypercomplex quotient construction      225
Hyperkahler manifold      54 100 201Ч219
Hyperkahler manifold, ALE      201 205Ч207
Hyperkahler manifold, cohomology      217
Hyperkahler manifold, examples      218Ч219
Hyperkahler manifold, Hodge numbers      216
Hyperkahler manifold, moduli space      217
Hyperkahler manifold, twistor space      204Ч205
Hyperkahler quotient construction      207 224
Hyperkahler structure      201 203
Hypersurface      98
Hypersurface, Calabi Ч Yau      140
Hypersurface, degree d      140
Hypersurface, in toric variety      143
Hypersurface, in weighted projective space      141Ч142
Immersion      65
Implicit Mapping Theorem      7
Injectivity radius      5 173 235 236 247
Instanton      225 253
Integrable systems      152Ч153 155 159 177 260Ч261
Integral affine structure      195 197 198
Integral current      72
Integral current, Cayley      277
Integral current, special Lagrangian      158 171 176
Interior estimate      15
Intrinsic torsion      38
Inverse mapping theorem      7 119
K3 surface      208Ч213
K3 surface, examples      208
K3 surface, marked      209
K3 surface, moduli space      209 213
Kahler chamber      210
Kahler class      84 88
Kahler cone      88 210 216
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