Differential geometry for physicists:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
1999
|
| Ausgabe: | Repr. |
| Schriftenreihe: | Advanced series on theoretical physical science
6 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 546 S. Ill., graph. Darst. |
| ISBN: | 9810231059 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV017614974 | ||
| 003 | DE-604 | ||
| 005 | 20080311 | ||
| 007 | t | ||
| 008 | 031031s1999 ad|| |||| 00||| eng d | ||
| 020 | |a 9810231059 |9 981-02-3105-9 | ||
| 035 | |a (OCoLC)60197571 | ||
| 035 | |a (DE-599)BVBBV017614974 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-355 |a DE-384 |a DE-19 | ||
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 100 | 1 | |a Hou, Bo-Yu |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential geometry for physicists |c Bo-Yu Hou ; Bo-Yuan Hou |
| 250 | |a Repr. | ||
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1999 | |
| 300 | |a XIII, 546 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advanced series on theoretical physical science |v 6 | |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Hou, Bo-Yuan |e Verfasser |4 aut | |
| 830 | 0 | |a Advanced series on theoretical physical science |v 6 |w (DE-604)BV012400065 |9 6 | |
| 856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010597170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-010597170 | ||
Datensatz im Suchindex
| _version_ | 1804130370003664896 |
|---|---|
| adam_text | Contents
Preface
χι
Differentiable
Manifolds and Differential Forms
1
1.1
Manifold
.................................. 1
1.2
Differentiable manifold
.......................... 7
1.3
Tangent space and tangent vector field
................. 15
1.4
Cotangent vector field
.......................... 20
1.5
Tensor product, exterior product and various higher order tensor fields
24
1.6
Exterior differentiation
........,................. 32
1.7
Orientation and Stokes formula
.................... 37
Notations and formulae
............................. 41
Exercises
..................................... 43
Transformation of Manifold, Manifolds with Given Vector Fields and
Lie Group Manifold
45
2.1
Continuous mapping between manifolds and its induced mapping
. . 45
2.2
Integral submanifold and Frobenius theorem
.............. 48
2.3
Integrability of differential equations and Frobenius theorem in terms
of differential forms
............................ 52
2.4
The flow of vector fields, one parameter local Lie transformation groups
and Lie derivative
............................. 58
2.5
Lie group, Lie algebra and exponential map
.............. 65
2.6
Lie transformation groups, orbit and the space of orbits
........ 73
Notations and Formulae
............................ 79
Exercises
..................................... 81
Affine
Connection and Covariant Differentiation
83
3.1
Moving frame approach to tensor field
................. 83
3.2 Affine
connection and covariant differentiation
............. 85
3.3
The curvature 2-form and the curvature tensor
.........: . . 91
3.4
Torsion tensor
............................... 93
3.5
Covariant exterior differential
...................... 98
3.6
Holonomy group of connections
..................... 101
V1 content
3.7
Berry phase, holonomy in physical system
............... 102
Notations and Formulea
............................ 105
Exercises
..................................... 106
4
Riemannian Manifold
107
4.1
Metric tensor field. Hodge star and codifferentiation
.......... 107
4.2
Riemannian connection
.........................
1]9
4.3
Riemannian curvature
.......................... 122
4.4
Bianchi
identity and Einstein field equation of gravity
......... 125
4.5
Isometry,
conformai
transformation and constant curvature space
. . 127
4.6
Orthogonal frame field and spin connection
............... 130
4.7
Surfaces and curves in 3-dimensional Euclidean space
......... 137
4.8
The computation of Riemannian curvature tensor
........... 148
4.9
Pseudosphere and
Backlund
transformation
.............. 154
Notations and Formulae
............................ 159
Exercises
..................................... 161
5
Symplectic Manifold and Contact Manifold
163
5.1
Symplectic. manifold
........................... 163
5.2
Special submanifolds of symplectic manifold
.............. 166
5.3
Symplectic and Hamiltonian vector fields.
Poisson
bracket
....... 168
5.4
Poission manifold and symplectic leaves
................. 170
5.5
Homogeneous symplectic manifold and the reduced phase space
... 173
5.6
Contact manifold
............................. 176
Notations and Formulae
.............................
ISO
Exercises
..................................... 181
6
Complex Manifolds
183
6.1
Complex structure of manifolds, almost complex manifolds
...... 183
6.2
Integrable
condition of almost complex structure
............ 190
6.3
Hermitian manifold
............................ 193
6.4 Kahler
manifold
.............................. 200
6.5
Connections on complex manifold
.................... 203
6.6
Riemannian symmetric space, its
Kahler
structure and nonlinear real¬
ization
................................... 209
6.7
Nonlinear <7-rnodels, soliton solutions and their geometric meaning
. . 216
Notations and Formulae
............................ 227
Exercises
..................................... 230
7
Homology of Manifolds
231
7.1
Homotopic mapping and manifolds with the same homotopy type
. . 232
7.2
Singular homology group
........................ 234
7.3
General homology group and universal coefficient theorem
...... 240
content
vii
7.4
Cohomology theory
............................ 246
7.5
de Rham
cohomology theory
....................... 249
7.6
Harmonic forms
.............................. 255
7.7
Bi-invariant form on group manifold and invariant form on symmetric
space
.................................... 257
7.8
G-structure of manifold and its restriction to the homology group of
manifold
.................................. 259
Notations and Formulae
............................ 261
Exercises
..................................... 262
8
Homotopy of Manifold, Fibre Bundle, Classification of Fibre Bun¬
dles
263
8.1
Homotopy group of manifold
....................... 263
8.2
Relative homotopy group and exact homotopy sequence
. . . 267
8.3
Relation between homotopy group and homology group
........ 275
8.4
Fibre bundle
................................ 278
8.5
Principal bundle and associated bundle
................ 284
8.6
Induced bundle, reduction of fibre bundle
................ 286
8.7
The homotopy classification of fibre bundles, universal fibre bundle
. 290
Notations and Formulae
............................ 294
Exercises
..................................... 295
9
Differential Geometry of Fibre Bundle, Yang-Mills Gauge Theory
297
9.1
Connection and curvature on principal bundle
............. 297
9.2
Connection on associated vector bundle
................. 303
9.3
Connection on general vector bundle
.................. 306
9.4
Gauge theory, action and Yang-Mills equation
............. 311
9.5
Local gauge symmetry and current conservation
............ 315
9.6
Instanton
................................. 319
9.7
Yang-Mills-Higgs
monopole
....................... 324
9.8
Seíberg-Witten
monopole
equation
................... 327
Notations and Formulae
............................ 330
Exercises
..................................... 332
10
Characteristic Classes
333
10.1
Introduction, Weil homomorphism
.................... 333
10.2
Chern class, the splitting principle
.................... 337
10.3
Pontrjagin class
.............................. 343
10.4
Euler
class
................................. 346
10.5 Stiefel-Whitney
class,
orientation and spin structure
...................... 347
10.6
Secondary characteristic class (Chern-Simons form)
.......... 351
10.7
Generalized Chern-Simons forms
..................... 357
V11
content
Notations and Formulae
............................ 360
Exercise
..................................... 362
11
The Atiyah-Singer Index Theorem
363
11.1
Introduction,
Euler
number and the associated theorem
........ 363
11.2
Elliptic differential operator, elliptic complex and its analytic index
. 365
11.3
Atiyah-Singer index theorem, the symbol bundle and its topological
index
.................................... 370
11.4
Other classical elliptic complex
..................... 373
11.5
Twisted elliptic complex
......................... 378
11.6
Brief comment on the proof for index theorem, the heat kernel method
381
11.7
Some applications in physics
....................... 386
Notations and Formulae
............................ 388
Exercise
..................................... 388
12
Index Theorem on Manifold with Boundary and on Open Infinite
Manifold
389
12.1
Introduction
................................ 389
12.2
Index theorem for
de
Rham complex on manifold with boundary
. . . 391
12.3
APS index theorem
............................ 392
12.4
APS index theorem for spin complex, spectral boundary condition
. . 395
12.5
Index theorem on open infinite manifold
................ 400
12.6
Weak local boundary condition for Dirac operator
........... 406
Notations and Formulae
............................ 410
Exercise
..................................... 410
13
Family Index Theorem, Topological properties of Quantum Gauge
Theory
411
13.1
Family index theorem of Dirac operator
................. 412
13.2
Relation among cohomology on orbit space, on connection space and
on gauge group
.............................. 415
13.3
Topological obstruction of variety degree on gauge group and the
Cech-
de Rham
double complex
......................... 418
13.4
The cocycle density of gauge group and gauge algebra
........ 425
13.5
Topological properties of 4-dim quantum
Yang-Mills theory and 0-vacuum
..................... 429
13.6
3-dim Yang-Mills theory and topological mass term
.......... 433
13.7
Fermion interaction and quantum anomalies
.............. 435
13.8
Topological interpretation of quantum anomalies
............ 441
Notations and Formulae
............................ 444
Exercise
..................................... 446
content
¡x
14
Noncommutative
Geometry,
Quantum
Group,
and g-deformation of
С
hern-Characters
447
14.1
Introduction
................................ 447
14.2
Linear transformations on the quantum
hyperplane,
quantum group
GLq{2) and SUq{2)
.................. .......... 449
14.3
Bicovariant calculus on quantum group SUq{2)
............. 452
14.4 Q-gauge
theory in terms of
ş-BRST
algebra
.............. 458
14.5
Q-Deformed Chern Class
......................... 461
14.6
q-Deformed Chern-Simons
........................ 462
14.7
q-Deformed Cocycle Hierarchy
...................... 465
Notations and Formulae
............................ 467
Exercise
..................................... 468
Appendix
A Simple Introduction to Set Theory
469
A.I Basic definitions and notations
...................... 469
A.
2
Equivalence relations and equivalence classes
.............. 470
A.3 Partial Ordering and Total Ordering
.................. 470
A.4 Maps
.................................... 471
В
Preliminary Topology
473
B.I Metric space
................................ 473
B.2 General
topologica!
space
......................... 474
B.3 Connectedness
............................... 475
B.4 Compactness
............................... 476
B.5 Product
.................................. 477
С
Some Basic Algebraic Structures
479
C.I Group, ring and field
........................... 479
C.2 Vector space, module and algebra
.................... 480
C.3 Euclidean space
.............................. 483
C.4 Normed algebra and C-algebra
..................... 483
C.5 Homogeneous space
............................ 483
D
Homomorphism of Algebraic Structure and Tensor Algebra
485
D.I Linear functions and dual spaces, dual linear map
........... 485
D.2 Bilinear functions and tensor products
................. 486
D.3 Direct sum and tensor algebra
...................... 488
D.4 Derivator algebras
............................. 488
E
Exact Sequence of Homomorphism
489
x
content
F Abelian Group
491
F.I Linear independence and rank
...................... 491
F.2
Finite generated groups
.......................... 492
G
Stokes Theorem
493
H Quarternion
and Milnor Exotic Spheres
497
H.I Division algebras
............................. 497
H.2 Quarternion
H
.............................. 498
H.3 Milnor exotic spheres
........................... 499
I Generalized
Kronecker
б
notation
501
1.1 Definition
................................. 501
1.2
Contractions of order-p
б
notation (p
<
n)
............... 501
1.3
Contraction of
б
and tensors
....................... 502
1.4
Generalized Levi-Civita notation
.................... 502
J
n-dimensional
Sphere Sn
505
K SU{2)
Group Manifold
507
L S2
Manifold and
Hopf
Maps
509
M
Clifford Algebra, Spin Group and its Representations
513
M.I Clifford Algebra
.............................. 513
M.2 Spin Group
................................ 514
M.3 The representation of
Spin(n)
...................... 515
M.
4
spin(n) algebra module
.......................... 517
N
Good Cover and the Nerve of a Cover.
Čech
Cohomology
519
О
Symmetric Trace
521
P
Cohomology with Compact Support and Cohomology on Fibre Bun¬
dle, Thorn Class
523
Q
A-theory, Topological
Invariance
of Elliptic Operator
527
Ŕ
Cohomology of Lie group and Lie algebra
529
S
Some Recursion Relations of Q-BRST Algebra
531
References
534
Index
538
|
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| author_facet | Hou, Bo-Yu Hou, Bo-Yuan |
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| author_sort | Hou, Bo-Yu |
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| bvnumber | BV017614974 |
| classification_rvk | SK 370 |
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| id | DE-604.BV017614974 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:19:57Z |
| institution | BVB |
| isbn | 9810231059 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010597170 |
| oclc_num | 60197571 |
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| owner | DE-355 DE-BY-UBR DE-384 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-384 DE-19 DE-BY-UBM |
| physical | XIII, 546 S. Ill., graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | World Scientific |
| record_format | marc |
| series | Advanced series on theoretical physical science |
| series2 | Advanced series on theoretical physical science |
| spelling | Hou, Bo-Yu Verfasser aut Differential geometry for physicists Bo-Yu Hou ; Bo-Yuan Hou Repr. Singapore [u.a.] World Scientific 1999 XIII, 546 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced series on theoretical physical science 6 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Hou, Bo-Yuan Verfasser aut Advanced series on theoretical physical science 6 (DE-604)BV012400065 6 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010597170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hou, Bo-Yu Hou, Bo-Yuan Differential geometry for physicists Advanced series on theoretical physical science Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Differential geometry for physicists |
| title_auth | Differential geometry for physicists |
| title_exact_search | Differential geometry for physicists |
| title_full | Differential geometry for physicists Bo-Yu Hou ; Bo-Yuan Hou |
| title_fullStr | Differential geometry for physicists Bo-Yu Hou ; Bo-Yuan Hou |
| title_full_unstemmed | Differential geometry for physicists Bo-Yu Hou ; Bo-Yuan Hou |
| title_short | Differential geometry for physicists |
| title_sort | differential geometry for physicists |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010597170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012400065 |
| work_keys_str_mv | AT houboyu differentialgeometryforphysicists AT houboyuan differentialgeometryforphysicists |