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Berline N., Getzler E., Vergne M. — Heat Kernels and Dirac Operators
Berline N., Getzler E., Vergne M. — Heat Kernels and Dirac Operators

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Название: Heat Kernels and Dirac Operators

Авторы: Berline N., Getzler E., Vergne M.

Аннотация:

In this book, the Atiyah-Singer index theorem for Dirac operators on compact Riemannian manifolds and its more recent generalizations receive simple proofs. The main technique which is used is an explicit geometric construction of the heat kernels of a generalized Dirac operator. The first four chapters could be used at the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. Several chapters deal with other preparatory material.


Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 369

Добавлена в каталог: 20.10.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L^{2}$-metric      302
$\bar{\partial}$-operator      136
$\hat{A}$-genus      50 144
$\hat{A}$-genus equivariant      213
$\mathcal{P}$-endomorphism      279 310
Adjoint, formal      68 89
Affine connection      28
Almost-complex manifold      135
Anomaly formula      350
Associated bundle      15
Atiyah — Bott fixed point formula      187
Atiyah — Segal — Singer fixed point formula      196
Atiyah, M.F.      147 150 187 196 245 295 339
Baum, P.      225
Berezin integral      42 54 106 109 146 196 243
Bianchi identity      44
Bismut Laplacian      258
Bismut superconnection      332 347
Bismut, J.M.      258 282 332 338 348
Bochner — Kodaira formula      140
Borel — Weil — Bott theorem      257
Bott class      241
Bott, R.      187 224 257
Bundle, associated      15
Bundle, Clifford      113
Bundle, equivariant      16 209
Bundle, frame      15 32
Bundle, holomorphic      136
Bundle, horizontal      21
Bundle, index      281 299
Bundle, principal      15
Canonical line bundle      140
Cartan subalgebra      229
Casimir operator      172
Character formula, Kirillov      254
Character formula, Weyl      253
Characteristic class      47
Characteristic class, $\hat{A}$-genus      50 151 337
Characteristic class, Chern character      49 286
Characteristic class, equivariant      212
Characteristic class, Euler class      58 148
Characteristic class, L-genus      150
Characteristic class, Todd genus      51 152
Characteristic number, Bott's formula      225
Cheeger, J.      225
Chern character      49 286
Chern character, equivariant      212
Chern character, relative      146
Chern character, relative, equivariant      250
Chern character, relative, localized      195
Chern — Weil homomorphism      47
Chern — Weil homomorphism, equivariant      236 294 352
Chern, S.S.      59
Chevalley, C.      108 233
Chirality operator      109
Clifford algebra      103
Clifford algebra, quantization map      104
Clifford algebra, supertrace      110
Clifford algebra, symbol map      104
Clifford bundle      113
Clifford connection      117 329
Clifford module      103 114 127 325
Clifford module, equivariant      186
Clifford module, twisted      114
Clifford superconnectton      117
Coadjoint orbit, Fourier transform      228
Cohomology, de Rham      18 128 296
Cohomology, Dolbeault      139
Cohomology, equivariant      209
Complex manifold      135
Connection      20
Connection form      21 26
Connection, affine      28
Connection, Clifford      117 329
Connection, invariant      26 210
Connection, Levi — Civita      33
Connection, on a principal bundle      21
Connection, torsion-free      28
Contraction      18 69
Covariant derivative      22
Covariant derivative, holomorphic      136
Covariant derivative, tensor product      25
Curvature      21 23 44 321
Curvature, cut-off function      84
Curvature, equivariant      210
Curvature, Ricci      34 130
Curvature, Riemannian      33
Curvature, scalar      34 126 134
Curvature, twisting      120
Curvature, twisting equivariant      250
De Rham cohomology      18 128 296
de Rham cohomology, compact supports      19
de Rham's theorem      128
Delta distribution      192
density      30
Density, half      68
Density, integral of      30
Density, Riemannian      34
Density, vertical      278
Determinant line bundle      41 306 348
Determinant line bundle, $L^{2}$-metric      306
Determinant line bundle, connection      309 350
Determinant line bundle, curvature      348
Determinant line bundle, Quillen metric      306
Determinant, zeta-function      302
Difference bundle      295
Differential form      17
Differential form, basic      20
Differential form, equivariant      208
Differential form, harmonic      128
Differential form, horizontal      20
Differential form, integral along the fibre      31
Differential form, self-dual      132
Differential form, volume      31
Differential operator      64
Differential operator, elliptic      65
Dirac operator      116
Dirac operator, classical      127 134
Dirac operator, index of      123
Dirac, P.A.M.      102
Divergence      68
Dolbeault cohomology      139
Dolbeault's theorem      139
Duhamel's formula      98 169
Duistermaat — Heckman theorem      226
Duistermaat, J.J.      226
Equivariant $\hat{A}$-genus      213 249
Equivariant, bundle      16 209
Equivariant, characteristic class      212
Equivariant, Chern character      212
Equivariant, Chern character, relative      192 250
Equivariant, Chern — Weil homomorphism      236 294 352
Equivariant, Clifford module      186
Equivariant, cohomology      209
Equivariant, curvature      210
Equivariant, differential form      208
Equivariant, Euler class      213 218
Equivariant, exterior differential      208
Equivariant, index      186 250 293
Equivariant, relative Chern character      250
Equivariant, superconnection      210
Equivariant, twisting curvature      250
Euler class      58
Euler class, equivariant      213 218
Euler number      59 129 149
Euler number, holomorphic vector bundle      139 152
Exponential map      35
Exponential map, Jacobian of      38 168
Exterior algebra      103
Exterior differential      17 29
Exterior differential, equivariant      208
Family      278
Family index theorem      338
Family of smoothing operators      314
Flag manifold      232 254
Fourier transform of orbit      224
Frame bundle      16
Frame bundle, orthonormal      32
Freed, D.A.      348
Fundamental one-form      28
Gauss — Bonnet — Chern theorem      59
Gaussian integral      54 72 173 178 223
Generalized section      74
Geodesic      35
Gilkey, P.B.      147 193
Gradient      34
Green operator      92 288
Hamiltonian vector field      214 227
Harish — Chandra      231
Harmonic form      128
harmonic oscillator      145 153
Harmonic spinor      134
Heat equation      75
Heat equation, formal solution      83
Heat kernel      72 75 310
Heat kernel, asymptotic expansion      87
Heckman, G.      226
Hilbert — Schmidt operator      87
Hlrzebruch, F.      150 152 245
Hodge star      131
Hodge's theorem      128 139
Holomorphic vector bundle      136
Horizontal bundle      21
Horizontal lift      21
INDEX      123 145
Index bundle      270 281 299
Index density      146
Index density, equivariant      196
Index density, family      338
Index theorem, equivariant      196 250
Index theorem, local      146 181
Index, equivariant      186 250 293
Index, family      see “Index bundle”
Injectivity radius      35
Jacobian of exponential map      38
K-theory      295
Kaehler manifold      137 150 257
Karamata, J.      94
Kernel      74
Kirillov character formula      253
Kirillov formula      249 295 351
Kirillov, A.A.      253
Kodaira vanishing theorem      141
Kostant, B.      255
Kuenneth theorem      129
L-genus      149
Laplace — Beltrami operator      128 133
Laplacian, generalized      65
Leibniz's rule      17 22 44 282
Levi — Civita connection      33
Licbnerowicz formula      126 135 166 334
Lichnecowicz, A.      134
Lie bracket      17
Lie derivative      16
Lie derivative, E. Cartan's homotopy formula      18
Lie superalgebra      39
Liouville form      226 231
Local index theorem      146 181
Local index theorem, equivariant      196 250
Local index theorem, family      338
Localization theorem      215 251
Manifold, almost-complex      135
Manifold, complex      135
Manifold, Hermitian      135
Manifold, Kaehler      137 150
Manifold, Riemannian      32
Manifold, spin      114 144
Manifold, symplectic      214
Mathai, V.      52 241
McKean — Singer formula      124 145
McKean — Singer formula, equivariant      186
McKean — Singer formula, family      299
McKean, H.      124
Mean curvature      321
Mehler's formula      145 154 265 344
Mellin transform      300
Minakshisundaram, S.      101
moment      211
Moment, Riemannian      213 249
Moment, symplectic      214
Moment, twisting      250
Normal coordinates      35 71
Orientation      31
Parallel transport      26
Parametrix      77
Patodi, V.K.      147
Pfaffian      42 224
Pleijel, A.      101
Poincare duality      132
Poincare lemma      52
Poincare — Hopf theorem      59 61 225
Polarization      109
Principal bundle      15
Principal bundle, geodesic distance      198
Principal bundle, heat kernel      175 200
Quantization map      104
Quillen metric      306 348
Quillen, D.      43 46 52 241 306
Radial vector field      35
Relative Chern character      146
Relative Chern character, equivariant      250
Relative Chern character, localized      195
Relative supertrace      113 147
Renormalized limit      302
Ricci curvature      34 130
Riemann — Roch theorem      152 245
Riemann — Roch — Hirzebruch theorem      152
Riemannian curvature      33
Riemannian manifold      32
Riemannian metric      32
Riemannian moment      213 249
Rossman, W.      234
Scalar curvature      34 134
Schwartz kernel theorem      74
Second fundamental form      320
Section      14
Section generalized      74
Segal, G.B.      196
Signature      133 150
Signature operator      133
Signature theorem      150
simplex      77
Singer, I.M.      124 147 150 196 295 339
Smoothing operator      74
Smoothing operator, family      314
Spin group      106
Spin structure      114
Spinor bundle      115
Spinor module      109
Spinor supertrace      110
Stationary phase      226
Superalgebra      38
Superalgebra, Lie      39
Superalgebra, supercommutative      39
Superbundle      39
Supercommutator      39
Superconnection      44 282
Superconnection, Bismut      318
Superconnection, Clifford      117
Superconnection, curvature      44
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