Applied differential geometry: a modern introduction
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| Format: | Book |
| Language: | English |
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New Jersey [u.a.]
World Scientific
2007
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 1253 - 1293 |
| Physical Description: | XXXIV, 1311 S. Ill., graph. Darst. |
| ISBN: | 9789812706140 9812706143 |
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| 020 | |a 9789812706140 |9 978-981-270-614-0 | ||
| 020 | |a 9812706143 |9 981-270-614-3 | ||
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| 100 | 1 | |a Ivancevic, Vladimir G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Applied differential geometry |b a modern introduction |c Vladimir G. Ivancevic ; Tijana T. Ivancevic |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2007 | |
| 300 | |a XXXIV, 1311 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 1253 - 1293 | ||
| 650 | 4 | |a Geometry, Differential |v Textbooks | |
| 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 Ivancevic, Tijana T. |e Verfasser |4 aut | |
| 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=016163291&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| adam_text | Contents
Preface
vii
Glossary of Frequently Used Symbols
xi
1.
Introduction
1
1.1
Manifolds and Related Geometrical Structures
...... 1
1.1.1
Geometrical Atlas
.................. 6
1.1.2
Topological Manifolds
................ 8
1.1.2.1
Topological manifolds without
boundary
................. 10
1.1.2.2
Topological manifolds with boundary
. 10
1.1.2.3
Properties of topological manifolds
... 10
1.1.3
Differentiable Manifolds
.............. 12
1.1.4
Tangent and Cotangent Bundles of Manifolds
. . 14
1.1.4.1
Tangent Bundle of a Smooth Manifold
. 14
1.1.4.2
Cotangent Bundle of a Smooth
Manifold
................. 15
1.1.4.3
Fibre-, Tensor-, and Jet-Bundles
... 15
1.1.5
Riemannian Manifolds: Configuration Spaces
for Lagrangian Mechanics
............. 16
1.1.5.1
Riemann Surfaces
............ 17
1.1.5.2
Riemannian Geometry
.......... 19
1.1.5.3
Application: Lagrangian Mechanics
. 22
1.1.5.4
Finsler manifolds
............. 25
1.1.6
Symplectic Manifolds: Phase-Spaces for
Hamiltonian Mechanics
............... 25
xviii
Applied
Differential
Geometry: A Modern Introduction
1.-1.7
Lie Groups
...................... 28
1.1.7.1
Application: Physical Examples of
Lie Groups
................ 30
1.1.8
Application: A Bird View on Modern Physics
. 31
1.1.8.1
Three Pillars of 20th Century Physics
. 31
1.1.8.2
String Theory in Plain English
.... 33
1.2
Application: Paradigm of Differential-Geometric
Modelling of Dynamical Systems
.............. 46
2.
Technical Preliminaries: Tensors, Actions and Functors
51
2.1
Tensors: Local Machinery of Differential Geometry
.... 51
2.1.1
Transformation of Coordinates and Elementary
Tensors
........................ 51
2.1.1.1
Transformation of Coordinates
..... 52
2.1.1.2
Scalar Invariants
............. 53
2.1.1.3
Vectors and Covectors
.......... 53
2.1.1.4
Second-Order Tensors
.......... 54
2.1.1.5
Higher-Order Tensors
.......... 56
2.1.1.6
Tensor Symmetry
............ 57
2.1.2
Euclidean Tensors
.................. 58
2.1.2.1
Basis Vectors and the Metric Tensor
inEn
................... 58
2.1.2.2
Tensor Products in En
.......... 59
2.1.3
Covariant Differentiation
.............. 60
2.1.3.1
Christoffel s Symbols
........... 60
2.1.3.2
Geodesies
................. 61
2.1.3.3
Covariant Derivative
........... 61
2.1.3.4
Covariant Form of Differential
Operators
................. 62
2.1.3.5
Absolute Derivative
........... 63
2.1.3.6 3D
Curve Geometry: Frenet-Serret
Formulae
................. 64
2.1.3.7
Mechanical Acceleration and Force
... 64
2.1.4
Application: Covariant Mechanics
........ 65
2.1.4.1
Riemannian Curvature Tensor
..... 70
2.1.4.2
Exterior Differential Forms
....... 71
2.1.4.3
The Covariant Force Law
........ 76
2.1.5
Application: Nonlinear Fluid Dynamics
.... 78
2.1.5.1
Continuity Equation
........... 78
Contents xix
2.1.5.2 Forces
Acting on a Fluid
........ 80
2.1.5.3
Constitutive and Dynamical Equations
81
2.1.5.4
Navier-Stokes Equations
........ 82
2.2
Actions: The Core Machinery of Modern Physics
..... 83
2.3
Functors: Global Machinery of Modern Mathematics
... 87
2.3.1
Maps
......................... 88
2.3.1.1
Notes from Set Theory
......... 88
2.3.1.2
Notes From Calculus
........... 89
2.3.1.3
Maps
................... 89
2.3.1.4
Algebra of Maps
............. 89
2.3.1.5
Compositions of Maps
.......... 90
2.3.1.6
The Chain Rule
............. 90
2.3.1.7
Integration and Change of Variables
. . 90
2.3.1.8
Notes from General Topology
...... 91
2.3.1.9
Topologicei
Space
............ 92
2.3.1.10
Homotopy
................. 93
2.3.1.11
Commutative Diagrams
......... 95
2.3.1.12
Groups and Related Algebraic
Structures
................. 97
2.3.2
Categories
...................... 102
2.3.3
Functors
....................... 105
2.3.4
Natural Transformations
.............. 108
2.3.4.1
Compositions of Natural
Transformations
............. 109
2.3.4.2
Dinatural
Transformations
....... 109
2.3.5
Limits and Colimits
.................
Ill
2.3.6
Adjunction
......................
Ill
2.3.7
Abelian Categorical Algebra
............ 113
2.3.8
n-Categories
.................... 116
2.3.8.1
Generalization of Small Categories
. . 117
2.3.8.2
Topological Structure of n—Categories
. 121
2.3.8.3
Homotopy Theory and Related
n—Categories
............... 121
2.3.8.4
Categorification
............. 123
2.3.9
Application: n—Categorical Framework for
Higher Gauge Fields
. ............... 124
2.3.10
Application: Natural Geometrical Structures
. 128
2.3.11
Ultimate Conceptual Machines: Weak
n—Categories
.................... 132
xx
Applied Differential Geometry: A Modern Introduction
3.
Applied Manifold Geometry
137
3.1
Introduction
......................... 137
3.1.1
Intuition behind Einstein s Geometrodynamics
. . 138
3.1.2
Einstein s Geometrodynamics in Brief
....... 142
3.2
Intuition Behind the Manifold Concept
.......... 143
3.3
Definition of a Differentiable Manifold
........... 145
3.4
Smooth Maps between Smooth Manifolds
......... 147
3.4.1
Intuition behind Topological Invariants of
Manifolds
...................... 148
3.5
(Co)Tangent Bundles of Smooth Manifolds
........ 150
3.5.1
Tangent Bundle and Lagrangian Dynamics
.... 150
3.5.1.1
Intuition behind a Tangent Bundle
. . . 150
3.5.1.2
Definition of a Tangent Bundle
..... 150
3.5.2
Cotangent Bundle and Hamiltonian Dynamics
. . 153
3.5.2.1
Definition of a Cotangent Bundle
.... 153
3.5.3
Application: Command/Control in Human-
Robot Interactions
................. 154
3.5.4
Application: Generalized Bidirectional
Associative Memory
................. 157
3.6
Tensor Fields on Smooth Manifolds
............ 163
3.6.1
Tensor Bundle
.................... 163
3.6.1.1
Pull-Back
and Push-Forward
...... 164
3.6.1.2
Dynamical Evolution and Flow
..... 165
3.6.1.3
Vector-Fields and Their Flows
..... 167
3.6.1.4
Vector-Fields on
M
........... 167
3.6.1.5
Integral Curves as Dynamical
Trajectories
................ 168
3.6.1.6
Dynamical Flows on
M
......... 172
3.6.1.7
Categories of ODEs
........... 173
3.6.2
Differential Forms on Smooth Manifolds
..... 174
3.6.2.1
1-Forms on
M
.............. 174
3.6.2.2
fc-Forms on
M
.............. 176
3.6.2.3
Exterior Differential Systems
...... 179
3.6.3
Exterior Derivative and (Co)Homology
...... 180
3.6.3.1
Intuition behind Cohomology
...... 182
3.6.3.2
Intuition behind Homology
....... 183
3.6.3.3
De Rham
Complex and Homotopy
Operators
................. 185
Contents xxi
3.6.3.4
Stokes
Theorem and de Rham
Cohomology ............... 186
3.6.3.5
Euler-Poincaré
Characteristics of
M
. . 188
3.6.3.6
Duality of Chains and Forms on
M
. . 188
3.6.3.7
Hodge Star Operator and Harmonic
Forms
................... 190
3.7
Lie Derivatives on Smooth Manifolds
............ 192
3.7.1
Lie Derivative Operating on Functions
...... 192
3.7.2
Lie Derivative of Vector Fields
........... 194
3.7.3
Time Derivative of the Evolution Operator
.... 197
3.7.4
Lie Derivative of Differential Forms
........ 197
3.7.5
Lie Derivative of Various Tensor Fields
...... 198
3.7.6
Application: Lie-Derivative
Neurodynamics
. . 200
3.7.7
Lie Algebras
..................... 202
3.8
Lie Groups and Associated Lie Algebras
.......... 202
3.8.1
Definition of a Lie Group
.............. 203
3.8.2
Actions of Lie Groups on Smooth Manifolds
. . . 207
3.8.3
Basic Dynamical Lie Groups
............ 210
3.8.3.1
Galilei Group
............... 210
3.8.3.2
General Linear Group
.......... 211
3.8.4
Application: Lie Groups in
Biodynamics
.... 212
3.8.4.1
Lie Groups of Joint Rotations
...... 212
3.8.4.2
Euclidean Groups of Total Joint
Motions
.................. 216
3.8.4.3
Group Structure of Biodynamical
Manifold
................. 221
3.8.5
Application: Dynamical Games on
SE
(η)
-Groups
................... 227
3.8.5.1
Configuration Models for Planar
Vehicles
.................. 227
3.8.5.2
Two-Vehicles Conflict Resolution
Manoeuvres
................ 228
3.8.5.3
Symplectic Reduction and Dynamical
Games on SE(2)
............. 230
3.8.5.4
Nash Solutions for
Multi-
Vehicle
Manoeuvres
................ 233
3.8.6
Classical Lie Theory
................ 235
3.8.6.1
Basic Tables of Lie Groups and their
Lie Algebras
............... 236
xxii
Applied
Differential
Geometry: A Modern
Introduction
3.8.6.2
Representations of Lie groups
...... 239
3.8.6.3
Root Systems and Dynkin Diagrams
. . 240
3.8.6.4
Simple and
Semisimple
Lie Groups
and Algebras
............... 245
3.9
Lie Symmetries and Prolongations on Manifolds
..... 247
3.9.1
Lie Symmetry Groups
................ 247
3.9.1.1
Exponentiation of Vector Fields on
M
. 247
3.9.1.2
Lie Symmetry Groups and General DEs
249
3.9.2
Prolongations
.................... 250
3.9.2.1
Prolongations of Functions
....... 250
3.9.2.2
Prolongations of Differential Equations
251
3.9.2.3
Prolongations of Group Actions
.... 252
3.9.2.4
Prolongations of Vector Fields
..... 253
3.9.2.5
General Prolongation Formula
...... 254
3.9.3
Generalized Lie Symmetries
............ 256
3.9.3.1
Noether Symmetries
........... 257
3.9.4
Application: Biophysical PDEs
......... 261
3.9.4.1
The Heat Equation
............ 261
3.9.4.2
The Korteveg-De
Vries
Equation
.... 262
3.9.5
Lie-Invariant Geometric Objects
.......... 262
3.9.5.1
Robot Kinematics
............ 262
3.9.5.2
Maurer-Cartan 1-Forms
......... 264
3.9.5.3
General Structure of
Integrable
One-Forms
................ 265
3.9.5.4
Lax
Integrable
Dynamical Systems
. . . 267
3.9.5.5
Application: Burgers Dynamical
System
.................. 268
3.10
Riemannian Manifolds and Their Applications
...... 271
3.10.1
Local Riemannian Geometry
............ 271
3.10.1.1
Riemannian Metric on
M
........ 272
3.10.1.2
Geodesies on
M
............. 277
3.10.1.3
Riemannian Curvature on
M
...... 278
3.10.2
Global Riemannian Geometry
........... 281
3.10.2.1
The Second Variation Formula
..... 281
3.10.2.2
Gauss-Bonnet Formula
......... 284
3.10.2.3
Ricci
Flow on
M
............. 285
3.10.2.4
Structure Equations on
M
....... 287
3.10.3
Application: Autonomous Lagrangian
Dynamics
...................... 289
Contents xxiii
3.10.3.1 Basis
of
Lagrangian Dynamics..... 289
3.10.3.2
Lagrange-Poincaré
Dynamics...... 290
ЗД0.4
Core
Application:
Search for
Quantum
Gravity .
292
3.10.4.1
What is Quantum Gravity?
....... 292
3.10.4.2
Main Approaches to Quantum Gravity
. 293
3.10.4.3
Traditional Approaches to Quantum
Gravity
.................. 300
3.10.4.4
New Approaches to Quantum Gravity
. 304
3.10.4.5
Black Hole Entropy
........... 310
3.10.5
Basics of Morse and (Co)Bordism Theories
.... 311
3.10.5.1
Morse Theory on Smooth Manifolds
. . 311
3.10.5.2
(Co)Bordism Theory on Smooth
Manifolds
................. 314
3.11
Finsler Manifolds and Their Applications
......... 316
3.11.1
Definition of a Finsler Manifold
.......... 316
3.11.2
Energy Functional, Variations and
Extrema
. . . 317
3.11.3
Application: Finsler-Lagrangian Field Theory
. 321
3.11.4
Riemann-Finsler Approach to Information
Geometry
...................... 323
3.11.4.1
Model Specification and Parameter
Estimation
................ 323
3.11.4.2
Model Evaluation and Testing
..... 324
3.11.4.3
Quantitative Criteria
.......... 324
3.11.4.4
Selection Among Different Models
... 327
3.11.4.5
Riemannian Geometry of Minimum
Description Length
............ 330
3.11.4.6
Finsler Approach to Information
Geometry
................. 333
3.12
Symplectic Manifolds and Their Applications
....... 335
3.12.1
Symplectic Algebra
................. 335
3.12.2
Symplectic Geometry
................ 336
3.12.3
Application: Autonomous Hamiltonian
Mechanics
...................... 338
3.12.3.1
Basics of Hamiltonian Mechanics
.... 338
3.12.3.2
Library of Basic Hamiltonian Systems
. 351
3.12.3.3
Hamilton-Poisson Mechanics
...... 361
3.12.3.4
Completely
Integrable
Hamiltonian
Systems
.................. 363
xxiv
Applied
Differential
Geometry: A Modern
Introduction
3.12.3.5
Momentum Map and Symplectic
Reduction
................. 372
3.12.4
Multisymplectic Geometry
............. 374
3.13
Application: Biodynamics-Robotics
........... 375
3.13.1
Muscle-Driven Hamiltonian
Biodynamics
..... 376
3.13.2
Hamiltonian-Poisson Biodynamical Systems
. . . 379
3.13.3
Lie-Poisson
Neurodynamics
Classifier
....... 383
3.13.4
Biodynamical Functors
............... 384
3.13.4.1
The Covariant Force Functor
...... 384
3.13.4.2
Lie-Lagrangian Biodynamical Functor
. 385
3.13.4.3
Lie-Hamiltonian Biodynamical Functor
391
3.13.5
Biodynamical Topology
............... 401
3.13.5.1
(Co)Chain Complexes in
Biodynamics
. 401
3.13.5.2
Morse Theory in
Biodynamics
..... 405
3.13.5.3
Hodge-De Rham Theory in
Biodynamics
............... 415
3.13.5.4
Lagrangian-Hamiltonian Duality
in
Biodynamics
.............. 419
3.14
Complex and
Kahler
Manifolds and Their
Applications
......................... 428
3.14.1
Complex Metrics: Hermitian and
Kahler..... 431
3.14.2
Calabi-Yau Manifolds
................ 436
3.14.3
Special Lagrangian Submanifolds
......... 437
3.14.4
Doibeault Cohomology and Hodge Numbers
. . . 438
3.15
Conformai Killing-Riemannian
Geometry
......... 441
3.15.1
Conformai
Killing Vector-Fields and Forms on
M
442
3.15.2
Conformai
Killing Tensors and Laplacian
Symmetry
...................... 443
3.15.3
Application: Killing Vector and Tensor Fields
in Mechanics
..................... 445
3.16
Application: Lax-Pair Tensors in Gravitation
...... 448
3.16.1
Lax-Pair Tensors
.................. 450
3.16.2
Geometrization of the 3-Particle Open
Toda
Lattice
........................ 452
• 3.16.2.1
Tensorial
Lax Representation
...... 453
3.16.3
4D Generalizations
................. 456
3.16.3.1
Case I
................... 456
3.16.3.2
Case II
.................. 457
3.16.3.3
Energy-Momentum Tensors
....... 457
Contents xxv
3.17
Applied Unorthodox Geometries
.............. 458
3.17.1
Noncommutative
Geometry
............ 458
3.17.1.1
Moyal Product and
Noncommutative
Algebra
.................. 458
3.17.1.2
Noncommutative
Space-Time Manifolds
459
3.17.1.3
Symmetries and Diffeomorphisms on
Deformed Spaces
............. 462
3.17.1.4
Deformed Diffeomorphisms
....... 465
3.17.1.5
Noncommutative
Space-Time Geometry
467
3.17.1.6
Star-Products and Expanded
Einstein-Hubert
Action
......... 470
3.17.2
Synthetic Differential Geometry
.......... 473
3.17.2.1
Distributions
............... 474
3.17.2.2
Synthetic Calculus in Euclidean Spaces
476
3.17.2.3
Spheres and Balls as Distributions
... 478
3.17.2.4
Stokes Theorem for Unit Sphere
.... 480
3.17.2.5
Time Derivatives of Expanding Spheres
481
3.17.2.6
The Wave Equation
........... 482
4.
Applied Bundle Geometry
485
4.1
Intuition Behind a Fibre Bundle
.............. 485
4.2
Definition of a Fibre Bundle
................ . 486
4.3
Vector and
Affine
Bundles
.................. 491
4.3.1
The Second Vector Bundle of the Manifold
M
. . 495
4.3.2
The Natural Vector Bundle
............. 496
4.3.3
Vertical Tangent and Cotangent Bundles
..... 498
4.3.3.1
Tangent and Cotangent Bundles
Revisited
................. 498
4.3.4
Affine
Bundles
.................... 500
4.4
Application: Semi-Riemannian Geometrical Mechanics
501
4.4.1
Vector-Fields and Connections
........... 501
4.4.2
Hamiltonian Structures on the Tangent Bundle
. 503
4.5
К
-Theory and Its Applications
.............. 508
4.5.1
Topological
iť-Theory
............... 508
4.5.1.1
Bott
Periodicity Theorem
........ 509
4.5.2
Algebraic
Я
-Theory................
510
4.5.3
Chern Classes and Chern Character
........ 511
4.5.4
Atiyah s View on K— Theory
............ 515
4.5.5
Atiyah-Singer Index Theorem
........... 518
xxvi
Applied
Differential
Geometry: A Modern
Introduction
4.5.6
The Infinite-Order Case
.............. 520
4.5.7
Twisted K-Theory and the Verlinde Algebra
. . 523
4.5.8
Application:
/ť-Theory
in String Theory
... 526
4.5.8.1
Classification of Ramond-Ramond
Fluxes
................... 526
4.5.8.2
Classification of D-Branes
....... 528
4.6
Principal Bundles
...................... 529
4.7
Distributions and Foliations on Manifolds
......... 533
4.8
Application: Nonholonomic Mechanics
......... 534
4.9
Application: Geometrical Nonlinear Control
...... 537
4.9.1
Introduction to Geometrical Nonlinear Control
. . 537
419.2
Feedback Linearization
............... 539
4.9.3
Nonlinear Controllability
.............. 547
4.9.4
Geometrical Control of Mechanical Systems
. . . 554
4.9.4.1
Abstract Control System
........ 554
4.9.4.2
Global Controllability of Linear Control
Systems
.................. 555
4.9.4.3
Local Controllability of
Affine
Control
Systems
.................. 555
4.9.4.4
Lagrangian Control Systems
...... 556
4.9.4.5
Lie-Adaptive Control
.......... 566
4.9.5
Hamiltonian Optimal Control and Maximum
Principle
....................... 567
4.9.5.1
Hamiltonian Control Systems
...... 567
4.9.5.2
Pontryagin s Maximum Principle
.... 570
4.9.5.3 Affine
Control Systems
......... 571
4.9.6
Brain-Like Control Functor in
Biodynamics
. . . 573
4.9.6.1
Functor Control Machine
........ 574
4.9.6.2
Spinal Control Level
........... 576
4.9.6.3
Cerebellar Control Level
......... 581
4.9.6.4
Cortical Control Level
.......... 584
4.9.6.5
Open Liouville
Neurodynamics
and
Biodynamical Self-Similarity
...... 587
4.9.7
Brain-Mind Functorial Machines
......... 594
4.9.7.1
Neurodynamical
2—
Functor
....... 594
4.9.7.2
Solitary Thought Nets and
Emerging Mind
............. 597
4.9.8
Geometrodynamics of Human Crowd
....... 602
4.9.8.1
Crowd Hypothesis
............ 603
Contents xxvii
4.9.8.2 Geometrodynamics
of Individual
Agents
................... 603
4.9.8.3
Collective Crowd Geometrodynamics
. . 605
4.10
M
ulti
vector-Fields and Tangent- Valued Forms
...... 606
4.11
Application: Geometrical Quantization
......... 614
4.11.1
Quantization of Hamiltonian Mechanics
...... 614
4.11.2
Quantization of Relativistic Hamiltonian
Mechanics
...................... 617
4.12
Symplectic Structures on Fiber Bundles
.......... 624
4.12.1
Hamiltonian Bundles
................ 625
4.12.1.1
Characterizing Hamiltonian Bundles
. . 625
4.12.1.2
Hamiltonian Structures
......... 626
4.12.1.3
Marked Hamiltonian Structures
..... 630
4.12.1.4
Stability
.................. 632
4.12.1.5
Cohomological Splitting
......... 632
4.12.1.6
Homological Action of Ham(M) on
M
634
4.12.1.7
General Symplectic Bundles
....... 636
4.12.1.8
Existence of Hamiltonian Structures
. . 637
4.12.1.9
Classification of Hamiltonian Structures
642
4.12.2
Properties of General Hamiltonian Bundles
. . . . 645
4.12.2.1
Stability
.................. 645
4.12.2.2
Functorial Properties
.......... 648
4.12.2.3
Splitting of Rational Cohomology
... 650
4.12.2.4
Hamiltonian Bundles and Gromov-
Witten Invariants
............ 654
4.12.2.5
Homotopy Reasons for Splitting
.... 659
4.12.2.6
Action of the Homology of (M) on
#»(M)
.................. 661
4.12.2.7
Cohomology of General Symplectic
Bundles
.................. 664
4.13
Clifford Algebras, Spinors and Penrose Twistors
..... 666
4.13.1
Clifford Algebras and Modules
........... 666
4.13.1.1
The Exterior Algebra
.......... 669
4.13.1.2
The Spin Group
............. 672
4.13.1.3
4D Case
................. . 672
4.13.2
Spinors
........................ 675
4.13.2.1
Basic Properties
............. 675
4.13.2.2
4D Case
.................. 677
4.13.2.3
(Anti)
Self Duality
............ 681
xxviii
Applied
Differential
Geometry: A Modern
Introduction
4.13.2.4
Herrai
tian
Structure on the
S
pinors
. . 686
4.13.2.5
Symplectic Structure on the Spinors
. . 689
4.13.3
Penrose Twistor Calculus
.............. 691
4.13.3.1
Penrose Index Formalism
........ 691
4.13.3.2
Twistor Calculus
............. 698
4.13.4
Application: Rovelli s Loop Quantum Gravity
. 701
4.13.4.1
Introduction to Loop Quantum Gravity
701
4.13.4.2
Formalism of Loop Quantum Gravity
. 708
4.13.4.3
Loop Algebra
............... 709
4.13.4.4
Loop Quantum Gravity
......... 711
4.13.4.5
Loop States and Spin Network States
. 712
4.13.4.6
Diagrammatic Representation of the
States
................... 715
4.13.4.7
Quantum Operators
........... 716
4.13.4.8
Loop v.s. Connection Representation
. 717
4.14
Application: Seiberg-
Witten
Monopole
Field Theory
. 718
4.14.1
SUSY Formalism
.................. 721
4.14.1.1 N = 2
Supersymmetry
.......... 721
4.14.1.2
ЛГ
= 2
Super-Action
........... 721
4.14.1.3
Spontaneous Symmetry-Breaking
. . . 723
4.14.1.4
Holomorphy and Duality
........ 724
4.14.1.5
The SW Prepotential
.......... 724
4.14.2
Clifford Actions, Dirac Operators and Spinor
Bundles
....................... 725
4.14.2.1
Clifford Algebras and Dirac Operators
. 727
4.14.2.2
Spin and
Spiile
Structures
....... 729
4.14.2.3
Spinor Bundles
.............. 730
4.14.2.4
The Gauge Group and Its Equations
. . 731
4.14.3
Original SW Low Energy Effective Field Action
. 732
4.14.4
QED With Matter
................. 735
4.14.5
QCD With Matter
................. 737
4.14.6
Duality
........................ 738
4.14.6.1
Witten s Formalism
........... 740
4.14.7
Structure of the Moduli Space
........... 746
4.14.7.1
Singularity at Infinity
.......... 746
4.14.7.2
Singularities at Strong Coupling
.... 747
4.14.7.3
Effects of a Massless
Monopole
..... 748
4.14.7.4
The Third Singularity
.......... 749
Contents
XXIX
4.14.7.5 Monopole
Condensation
and
Confinement
............... 750
4.14.8
Masses and Periods
................. 752
4.14.9
Residues
....................... 754
4.14.10
SW
Monopole
Equations and Donaldson Theory
. 757
4.14.10.1
Topologica!
Invariance..........
760
4.14.10.2
Vanishing Theorems
........... 762
4.14.10.3
Computation on
Kahler
Manifolds
. . . 765
4.14.11
SW Theory and
Integrable
Systems
........ 769
4.14.11.1
SU(N) Elliptic CM System
....... 771
4.14.11.2
CM Systems Defined by Lie Algebras
. 772
4.14.11.3
Twisted CM-Systems Defined by
Lie Algebras
............... 773
4.14.11.4
Scaling Limits of CM-Systems
..... 774
4.14.11.5
Lax Pairs for CM-Systems
....... 776
4.14.11.6
CM and SW Theory for SU(N)
.... 779
4.14.11.7
CM and SW Theory for General
Lie Algebra
................ 782
4.14.12
SW Theory and WDW Equations
........ 784
4.14.12.1
WDW Equations
............ 784
4.14.12.2
Perturbative SW Prepotentials
..... 787
4.14.12.3
Associativity Conditions
......... 789
4.14.12.4
SW Theories and
Integrable
Systems
. . 790
4.14.12.5
WDW Equations in SW Theories
... 793
5.
Applied Jet Geometry
797
5.1
Intuition Behind a Jet Space
................ 797
5.2
Definition of a 1-Jet Space
................. 801
5.3
Connections as Jet Fields
.................. 806
5.3.1
Principal Connections
................ 815
5.4
Definition of a 2-Jet Space
................. 818
5.5
Higher-Order Jet Spaces
.................. 822
5.6
Application: Jets and
Non-
Autonomous Dynamics
. . . 824
5.6.1
Geodesies
...................... 830
5.6.2
Quadratic Dynamical Equations
.......... 831
5.6.3
Equation of Free-Motion
.............. 832
5.6.4
Quadratic Lagrangian and Newtonian Systems
. . 833
5.6.5
Jacobi Fields
..................... 835
5.6.6
Constraints
...................... 836
xxx Applied Differential
Geometry: A Modem
Introduction
5.6.7
Time-Dependent Lagrangian Dynamics
...... 841
516.8
Time-Dependent Hamiltonian Dynamics
..... 843
5.6.9
Time-Dependent Constraints
........... 848
5.6.10
Lagrangian Constraints
............... 849
5.6.11
Quadratic Degenerate Lagrangian Systems
.... 852
5.6.12
Time—Dependent
Integrable
Hamiltonian Systems
855
5.6.13
Time-Dependent Action-Angle Coordinates
. . . 858
5.6.14
Lyapunov Stability
................. 860
5.6.15
First-Order Dynamical Equations
......... 861
5.6.16
Lyapunov Tensor and Stability
........... 863
5.6.16.1
Lyapunov Tensor
............. 863
5.6.16.2
Lyapunov Stability
............ 864
5.7
Application: Jets and Multi-Time Rheonomic
Dynamics
........................... 868
5.7.1
Relativistic Rheonomic Lagrangian Spaces
.... 870
5.7.2
Canonical Nonlinear Connections
......... 871
5.7.3
Cartan s Canonical Connections
.......... 874
5.7.4
General Nonlinear Connections
.......... 876
5.8
Jets and Action Principles
.................. 877
5.9
Application: Jets and Lagrangian Field Theory
..... 883
5.9.1
Lagrangian Conservation Laws
........... 888
5.9.2
General Covariance Condition
........... 893
5.10
Application: Jets and Hamiltonian Field Theory
.... 897
5.10.1
Covariant Hamiltonian Field Systems
....... 899
5.10.2
Associated Lagrangian and Hamiltonian Systems
902
5.10.3
Evolution Operator
................. 904
5.10.4
Quadratic Degenerate Systems
........... 910
5.11
Application: Gauge Fields on Principal Connections
. . 913
5.11.1
Connection Strength
................ 913
5.11.2
Associated Bundles
................. 914
5.11.3
Classical Gauge Fields
............... 915
5.11.4
Gauge Transformations
............... 917
5.11.5
Lagrangian Gauge Theory
............. 919
5.11.6
Hamiltonian Gauge Theory
............. 920
5.11.7
Gauge Conservation Laws
............. 923
5.11.8
Topologica!
Gauge Theories
............ 924
5.12
Application: Modern Geometrodynamics
........ 928
5.12.1
Stress-Energy-Momentum Tensors
........ 928
5.12.2
Gauge Systems of Gravity and Fermion Fields
. . 955
Contents xxxi
5.12.3 Hawking-Penrose Quantum
Gravity and
Black
Holes
..................... 963
6.
Geometrical Path
Integrals
and Their Applications
983
6.1
Intuition Behind a Path Integral
.............. 984
6.1.1
Classical Probability Concept
........... 984
6.1.2
Discrete Random Variable
............. 984
6.1.3
Continuous Random Variable
........... 984
6.1.4
General Markov Stochastic Dynamics
....... 985
6.1.5
Quantum Probability Concept
........... 989
6.1.6
Quantum Coherent States
............. 991
6.1.7
Dirac s <bra ket> Transition Amplitude
. ... 992
6.1.8
Feynman s Sum-over-Histories
........... 994
6.1.9
The Basic Form of a Path Integral
......... 996
6.1.10
Application: Adaptive Path Integral
...... 997
6.2
Path Integral History
.................... 998
6.2.1
Extract from Feynman s Nobel Lecture
...... 998
6.2.2
Lagrangian Path Integral
.............. 1002
6.2.3
Hamiltonian Path Integral
............. 1003
6.2.4
Feynman-Kac Formula
............... 1004
6.2.5
Ito
Formula
..................... 1006
6.3
Standard Path-Integral Quantization
........... 1006
6.3.1
Canonical versus Path-Integral Quantization
. . . 1006
6.3.2
Application: Particles, Sources, Fields and
Gauges
........................ 1011
6.3.2.1
Particles
.................. 1011
6.3.2.2
Sources
.................. 1012
6.3.2.3
Fields
................... 1013
6.3.2.4
Gauges
.................. 1013
6.3.3
Riemannian-Symplectic Geometries
........ 1014
6.3.4
Euclidean Stochastic Path Integral
........ 1016
6.3.5
Application: Stochastic Optimal Control
.... 1020
6.3.5.1
Path-Integral Formalism
........ 1021
6.3.5.2
Monte Carlo Sampling
.......... 1023
6.3.6
Application: Nonlinear Dynamics of Option
Pricing
........................ 1025
6.3.6.1
Theory and Simulations of Option
Pricing
.................. 1025
6.3.6.2
Option Pricing via Path Integrals
. . . 1029
xxxii
Applied
Differential
Geometry: A Modern
Introduction
б.З.б.З
Continuum Limit and American
Options
.................. 1035
6.3.7
Application: Nonlinear Dynamics of Complex
Nets
......................... 1036
6.3.7.1
Continuum Limit of the Kuramoto Net
1037
6.3.7.2
Path-Integral Approach to Complex
Nets
.................... 1038
6.3.8
Application: Dissipative Quantum Brain Model
1039
6.3.9
Application: Cerebellum as a Neural
Path-Integral
.................... 1043
6.3.9.1
Spinal Autogenetic Reflex Control
, . . 1045
6.3.9.2
Cerebellum
-
the Comparator
..... 1047
6.3.9.3
Hamiltonian Action and Neural Path
Integral
.................. 1049
6.3.10
Path Integrals via Jets: Perturbative Quantum
Fields
......................... 1050
6.4
Sum over Geometries and Topologies
............ 1055
6.4.1
Simplicial Quantum Geometry
........... 1057
6.4.2
Discrete Gravitational Path Integrals
....... 1059
6.4.3
Regge
Calculus
................... 1061
6.4.4
Lorentzian Path Integral
.............. 1064
6.4.5
Application: Topological Phase Transitions
and Hamiltonian Chaos
............... 1069
6.4.5.1
Phase Transitions in Hamiltonian
Systems
.................. 1069
6.4.5.2
Geometry of the Largest Lyapunov
Exponent
................. 1072
6.4.5.3
Euler
Characteristics of Hamiltonian
Systems
.................. 1075
6.4.6
Application: Force-Field Psychodynamics
. . . 1079
6.4.6.1
Motivational Cognition in the Life
Space Foam
................ 1080
6.5
Application: Witten s TQFT, SW-Monopoles and
Strings
............................. 1097
6.5.1
Topological Quantum Field Theory
........ 1097
6.5.2
Seiberg-Witten Theory and TQFT
........ 1103
6.5.2.1
S
W
Invariants and
Monopole
Equations
1103
6.5.2.2
Topological Lagrangian
......... 1105
6.5.2.3
Quantum Field Theory
......... 1107
Contents xxxiii
6.5.2.4
Dimensional
Reduction and
3D
Field
Theory
.................. 1112
6.5.2.5
Geometrical Interpretation
....... 1115
6.5.3
TQFTs Associated with SW-Monopoles
..... 1118
6.5.3.1
Dimensional Reduction
......... 1122
6.5.3.2
TQFTs of
3D
Monopoles
........ 1124
6.5.3.3
Non-Abelian Case
............ 1135
6.5.4
Stringy Actions and Amplitudes
.......... 1138
6.5.4.1
Strings
.................. 1139
6.5.4.2
Interactions
................ 1140
6.5.4.3
Loop Expansion
-
Topology of Closed
Surfaces
.................. 1141
6.5.5
Transition Amplitudes for Strings
......... 1143
6.5.6
Weyl
Invariance
and Vertex Operator Formulation
1146
6.5.7
More General Actions
................ 1146
6.5.8
Transition Amplitude for a Single Point Particle
. 1147
6.5.9
Witten s Open String Field Theory
........ 1148
6.5.9.1
Operator Formulation of String Field
Theory
.................. 1149
6.5.9.2
Open Strings in Constant B—Field
Background
................ 1151
6.5.9.3
Construction of Overlap Vertices
.... 1154
6.5.9.4
Transformation of String Fields
..... 1164
6.6
Application: Dynamics of Strings and Branes
...... 1168
6.6.1
A Relativistic Particle
............... 1169
6.6.2
A String
....................... 1171
6.6.3
A Brane
....................... 1173
6.6.4
String Dynamics
................... 1175
6.6.5
Brane
Dynamics
................... 1177
6.7
Application:
Topologica!
String Theory
......... 1180
6.7.1
Quantum Geometry Framework
.......... 1180
6.7.2
Green-Schwarz Bosonic Strings and Branes
.... 1181
6.7.3
Calabi-Yau Manifolds, Orbifolds and Mirror
Symmetry
...................... 1186
6.7.4
More on Topological Field Theories
........ 1189
6.7.5
Topological Strings
................. 1204
6.7.6
Geometrical Transitions
.............. 1221
6.7.7
Topological Strings and Black Hole Attractors
. . 1225
xxxiv
Applied Differential Geometry: A Modern Introduction
6.8
APPLICATION: Advanced Geometry and Topology of
String Theory
.......................... 1232
6.8.1
String Theory and
Noncommutative
Geometry
. . 1232
6.8.1.1
Noncommutative
Gauge Theory
.... 1233
6.8.1.2
Open Strings in the Presence of
Constant B-Field
............ 1235
6.8.2
K-Theory Classification of Strings
........ 1241
Bibliography
1253
Index
1295
|
| adam_txt |
Contents
Preface
vii
Glossary of Frequently Used Symbols
xi
1.
Introduction
1
1.1
Manifolds and Related Geometrical Structures
. 1
1.1.1
Geometrical Atlas
. 6
1.1.2
Topological Manifolds
. 8
1.1.2.1
Topological manifolds without
boundary
. 10
1.1.2.2
Topological manifolds with boundary
. 10
1.1.2.3
Properties of topological manifolds
. 10
1.1.3
Differentiable Manifolds
. 12
1.1.4
Tangent and Cotangent Bundles of Manifolds
. . 14
1.1.4.1
Tangent Bundle of a Smooth Manifold
. 14
1.1.4.2
Cotangent Bundle of a Smooth
Manifold
. 15
1.1.4.3
Fibre-, Tensor-, and Jet-Bundles
. 15
1.1.5
Riemannian Manifolds: Configuration Spaces
for Lagrangian Mechanics
. 16
1.1.5.1
Riemann Surfaces
. 17
1.1.5.2
Riemannian Geometry
. 19
1.1.5.3
Application: Lagrangian Mechanics
. 22
1.1.5.4
Finsler manifolds
. 25
1.1.6
Symplectic Manifolds: Phase-Spaces for
Hamiltonian Mechanics
. 25
xviii
Applied
Differential
Geometry: A Modern Introduction
1.-1.7
Lie Groups
. 28
1.1.7.1
Application: Physical Examples of
Lie Groups
. 30
1.1.8
Application: A Bird View on Modern Physics
. 31
1.1.8.1
Three Pillars of 20th Century Physics
. 31
1.1.8.2
String Theory in 'Plain English'
. 33
1.2
Application: Paradigm of Differential-Geometric
Modelling of Dynamical Systems
. 46
2.
Technical Preliminaries: Tensors, Actions and Functors
51
2.1
Tensors: Local Machinery of Differential Geometry
. 51
2.1.1
Transformation of Coordinates and Elementary
Tensors
. 51
2.1.1.1
Transformation of Coordinates
. 52
2.1.1.2
Scalar Invariants
. 53
2.1.1.3
Vectors and Covectors
. 53
2.1.1.4
Second-Order Tensors
. 54
2.1.1.5
Higher-Order Tensors
. 56
2.1.1.6
Tensor Symmetry
. 57
2.1.2
Euclidean Tensors
. 58
2.1.2.1
Basis Vectors and the Metric Tensor
inEn
. 58
2.1.2.2
Tensor Products in En
. 59
2.1.3
Covariant Differentiation
. 60
2.1.3.1
Christoffel's Symbols
. 60
2.1.3.2
Geodesies
. 61
2.1.3.3
Covariant Derivative
. 61
2.1.3.4
Covariant Form of Differential
Operators
. 62
2.1.3.5
Absolute Derivative
. 63
2.1.3.6 3D
Curve Geometry: Frenet-Serret
Formulae
. 64
2.1.3.7
Mechanical Acceleration and Force
. 64
2.1.4
Application: Covariant Mechanics
. 65
2.1.4.1
Riemannian Curvature Tensor
. 70
2.1.4.2
Exterior Differential Forms
. 71
2.1.4.3
The Covariant Force Law
. 76
2.1.5
Application: Nonlinear Fluid Dynamics
. 78
2.1.5.1
Continuity Equation
. 78
Contents xix
2.1.5.2 Forces
Acting on a Fluid
. 80
2.1.5.3
Constitutive and Dynamical Equations
81
2.1.5.4
Navier-Stokes Equations
. 82
2.2
Actions: The Core Machinery of Modern Physics
. 83
2.3
Functors: Global Machinery of Modern Mathematics
. 87
2.3.1
Maps
. 88
2.3.1.1
Notes from Set Theory
. 88
2.3.1.2
Notes From Calculus
. 89
2.3.1.3
Maps
. 89
2.3.1.4
Algebra of Maps
. 89
2.3.1.5
Compositions of Maps
. 90
2.3.1.6
The Chain Rule
. 90
2.3.1.7
Integration and Change of Variables
. . 90
2.3.1.8
Notes from General Topology
. 91
2.3.1.9
Topologicei
Space
. 92
2.3.1.10
Homotopy
. 93
2.3.1.11
Commutative Diagrams
. 95
2.3.1.12
Groups and Related Algebraic
Structures
. 97
2.3.2
Categories
. 102
2.3.3
Functors
. 105
2.3.4
Natural Transformations
. 108
2.3.4.1
Compositions of Natural
Transformations
. 109
2.3.4.2
Dinatural
Transformations
. 109
2.3.5
Limits and Colimits
.
Ill
2.3.6
Adjunction
.
Ill
2.3.7
Abelian Categorical Algebra
. 113
2.3.8
n-Categories
. 116
2.3.8.1
Generalization of'Small'Categories
. . 117
2.3.8.2
Topological Structure of n—Categories
. 121
2.3.8.3
Homotopy Theory and Related
n—Categories
. 121
2.3.8.4
Categorification
. 123
2.3.9
Application: n—Categorical Framework for
Higher Gauge Fields
. . 124
2.3.10
Application: Natural Geometrical Structures
. 128
2.3.11
Ultimate Conceptual Machines: Weak
n—Categories
. 132
xx
Applied Differential Geometry: A Modern Introduction
3.
Applied Manifold Geometry
137
3.1
Introduction
. 137
3.1.1
Intuition behind Einstein's Geometrodynamics
. . 138
3.1.2
Einstein's Geometrodynamics in Brief
. 142
3.2
Intuition Behind the Manifold Concept
. 143
3.3
Definition of a Differentiable Manifold
. 145
3.4
Smooth Maps between Smooth Manifolds
. 147
3.4.1
Intuition behind Topological Invariants of
Manifolds
. 148
3.5
(Co)Tangent Bundles of Smooth Manifolds
. 150
3.5.1
Tangent Bundle and Lagrangian Dynamics
. 150
3.5.1.1
Intuition behind a Tangent Bundle
. . . 150
3.5.1.2
Definition of a Tangent Bundle
. 150
3.5.2
Cotangent Bundle and Hamiltonian Dynamics
. . 153
3.5.2.1
Definition of a Cotangent Bundle
. 153
3.5.3
Application: Command/Control in Human-
Robot Interactions
. 154
3.5.4
Application: Generalized Bidirectional
Associative Memory
. 157
3.6
Tensor Fields on Smooth Manifolds
. 163
3.6.1
Tensor Bundle
. 163
3.6.1.1
Pull-Back
and Push-Forward
. 164
3.6.1.2
Dynamical Evolution and Flow
. 165
3.6.1.3
Vector-Fields and Their Flows
. 167
3.6.1.4
Vector-Fields on
M
. 167
3.6.1.5
Integral Curves as Dynamical
Trajectories
. 168
3.6.1.6
Dynamical Flows on
M
. 172
3.6.1.7
Categories of ODEs
. 173
3.6.2
Differential Forms on Smooth Manifolds
. 174
3.6.2.1
1-Forms on
M
. 174
3.6.2.2
fc-Forms on
M
. 176
3.6.2.3
Exterior Differential Systems
. 179
3.6.3
Exterior Derivative and (Co)Homology
. 180
3.6.3.1
Intuition behind Cohomology
. 182
3.6.3.2
Intuition behind Homology
. 183
3.6.3.3
De Rham
Complex and Homotopy
Operators
. 185
Contents xxi
3.6.3.4
Stokes
Theorem and de Rham
Cohomology . 186
3.6.3.5
Euler-Poincaré
Characteristics of
M
. . 188
3.6.3.6
Duality of Chains and Forms on
M
. . 188
3.6.3.7
Hodge Star Operator and Harmonic
Forms
. 190
3.7
Lie Derivatives on Smooth Manifolds
. 192
3.7.1
Lie Derivative Operating on Functions
. 192
3.7.2
Lie Derivative of Vector Fields
. 194
3.7.3
Time Derivative of the Evolution Operator
. 197
3.7.4
Lie Derivative of Differential Forms
. 197
3.7.5
Lie Derivative of Various Tensor Fields
. 198
3.7.6
Application: Lie-Derivative
Neurodynamics
. . 200
3.7.7
Lie Algebras
. 202
3.8
Lie Groups and Associated Lie Algebras
. 202
3.8.1
Definition of a Lie Group
. 203
3.8.2
Actions of Lie Groups on Smooth Manifolds
. . . 207
3.8.3
Basic Dynamical Lie Groups
. 210
3.8.3.1
Galilei Group
. 210
3.8.3.2
General Linear Group
. 211
3.8.4
Application: Lie Groups in
Biodynamics
. 212
3.8.4.1
Lie Groups of Joint Rotations
. 212
3.8.4.2
Euclidean Groups of Total Joint
Motions
. 216
3.8.4.3
Group Structure of Biodynamical
Manifold
. 221
3.8.5
Application: Dynamical Games on
SE
(η)
-Groups
. 227
3.8.5.1
Configuration Models for Planar
Vehicles
. 227
3.8.5.2
Two-Vehicles Conflict Resolution
Manoeuvres
. 228
3.8.5.3
Symplectic Reduction and Dynamical
Games on SE(2)
. 230
3.8.5.4
Nash Solutions for
Multi-
Vehicle
Manoeuvres
. 233
3.8.6
Classical Lie Theory
. 235
3.8.6.1
Basic Tables of Lie Groups and their
Lie Algebras
. 236
xxii
Applied
Differential
Geometry: A Modern
Introduction
3.8.6.2
Representations of Lie groups
. 239
3.8.6.3
Root Systems and Dynkin Diagrams
. . 240
3.8.6.4
Simple and
Semisimple
Lie Groups
and Algebras
. 245
3.9
Lie Symmetries and Prolongations on Manifolds
. 247
3.9.1
Lie Symmetry Groups
. 247
3.9.1.1
Exponentiation of Vector Fields on
M
. 247
3.9.1.2
Lie Symmetry Groups and General DEs
249
3.9.2
Prolongations
. 250
3.9.2.1
Prolongations of Functions
. 250
3.9.2.2
Prolongations of Differential Equations
251
3.9.2.3
Prolongations of Group Actions
. 252
3.9.2.4
Prolongations of Vector Fields
. 253
3.9.2.5
General Prolongation Formula
. 254
3.9.3
Generalized Lie Symmetries
. 256
3.9.3.1
Noether Symmetries
. 257
3.9.4
Application: Biophysical PDEs
. 261
3.9.4.1
The Heat Equation
. 261
3.9.4.2
The Korteveg-De
Vries
Equation
. 262
3.9.5
Lie-Invariant Geometric Objects
. 262
3.9.5.1
Robot Kinematics
. 262
3.9.5.2
Maurer-Cartan 1-Forms
. 264
3.9.5.3
General Structure of
Integrable
One-Forms
. 265
3.9.5.4
Lax
Integrable
Dynamical Systems
. . . 267
3.9.5.5
Application: Burgers Dynamical
System
. 268
3.10
Riemannian Manifolds and Their Applications
. 271
3.10.1
Local Riemannian Geometry
. 271
3.10.1.1
Riemannian Metric on
M
. 272
3.10.1.2
Geodesies on
M
. 277
3.10.1.3
Riemannian Curvature on
M
. 278
3.10.2
Global Riemannian Geometry
. 281
3.10.2.1
The Second Variation Formula
. 281
3.10.2.2
Gauss-Bonnet Formula
. 284
3.10.2.3
Ricci
Flow on
M
. 285
3.10.2.4
Structure Equations on
M
. 287
3.10.3
Application: Autonomous Lagrangian
Dynamics
. 289
Contents xxiii
3.10.3.1 Basis
of
Lagrangian Dynamics. 289
3.10.3.2
Lagrange-Poincaré
Dynamics. 290
ЗД0.4
Core
Application:
Search for
Quantum
Gravity .
292
3.10.4.1
What is Quantum Gravity?
. 292
3.10.4.2
Main Approaches to Quantum Gravity
. 293
3.10.4.3
Traditional Approaches to Quantum
Gravity
. 300
3.10.4.4
New Approaches to Quantum Gravity
. 304
3.10.4.5
Black Hole Entropy
. 310
3.10.5
Basics of Morse and (Co)Bordism Theories
. 311
3.10.5.1
Morse Theory on Smooth Manifolds
. . 311
3.10.5.2
(Co)Bordism Theory on Smooth
Manifolds
. 314
3.11
Finsler Manifolds and Their Applications
. 316
3.11.1
Definition of a Finsler Manifold
. 316
3.11.2
Energy Functional, Variations and
Extrema
. . . 317
3.11.3
Application: Finsler-Lagrangian Field Theory
. 321
3.11.4
Riemann-Finsler Approach to Information
Geometry
. 323
3.11.4.1
Model Specification and Parameter
Estimation
. 323
3.11.4.2
Model Evaluation and Testing
. 324
3.11.4.3
Quantitative Criteria
. 324
3.11.4.4
Selection Among Different Models
. 327
3.11.4.5
Riemannian Geometry of Minimum
Description Length
. 330
3.11.4.6
Finsler Approach to Information
Geometry
. 333
3.12
Symplectic Manifolds and Their Applications
. 335
3.12.1
Symplectic Algebra
. 335
3.12.2
Symplectic Geometry
. 336
3.12.3
Application: Autonomous Hamiltonian
Mechanics
. 338
3.12.3.1
Basics of Hamiltonian Mechanics
. 338
3.12.3.2
Library of Basic Hamiltonian Systems
. 351
3.12.3.3
Hamilton-Poisson Mechanics
. 361
3.12.3.4
Completely
Integrable
Hamiltonian
Systems
. 363
xxiv
Applied
Differential
Geometry: A Modern
Introduction
3.12.3.5
Momentum Map and Symplectic
Reduction
. 372
3.12.4
Multisymplectic Geometry
. 374
3.13
Application: Biodynamics-Robotics
. 375
3.13.1
Muscle-Driven Hamiltonian
Biodynamics
. 376
3.13.2
Hamiltonian-Poisson Biodynamical Systems
. . . 379
3.13.3
Lie-Poisson
Neurodynamics
Classifier
. 383
3.13.4
Biodynamical Functors
. 384
3.13.4.1
The Covariant Force Functor
. 384
3.13.4.2
Lie-Lagrangian Biodynamical Functor
. 385
3.13.4.3
Lie-Hamiltonian Biodynamical Functor
391
3.13.5
Biodynamical Topology
. 401
3.13.5.1
(Co)Chain Complexes in
Biodynamics
. 401
3.13.5.2
Morse Theory in
Biodynamics
. 405
3.13.5.3
Hodge-De Rham Theory in
Biodynamics
. 415
3.13.5.4
Lagrangian-Hamiltonian Duality
in
Biodynamics
. 419
3.14
Complex and
Kahler
Manifolds and Their
Applications
. 428
3.14.1
Complex Metrics: Hermitian and
Kahler. 431
3.14.2
Calabi-Yau Manifolds
. 436
3.14.3
Special Lagrangian Submanifolds
. 437
3.14.4
Doibeault Cohomology and Hodge Numbers
. . . 438
3.15
Conformai Killing-Riemannian
Geometry
. 441
3.15.1
Conformai
Killing Vector-Fields and Forms on
M
442
3.15.2
Conformai
Killing Tensors and Laplacian
Symmetry
. 443
3.15.3
Application: Killing Vector and Tensor Fields
in Mechanics
. 445
3.16
Application: Lax-Pair Tensors in Gravitation
. 448
3.16.1
Lax-Pair Tensors
. 450
3.16.2
Geometrization of the 3-Particle Open
Toda
Lattice
. 452
• 3.16.2.1
Tensorial
Lax Representation
. 453
3.16.3
4D Generalizations
. 456
3.16.3.1
Case I
. 456
3.16.3.2
Case II
. 457
3.16.3.3
Energy-Momentum Tensors
. 457
Contents xxv
3.17
Applied Unorthodox Geometries
. 458
3.17.1
Noncommutative
Geometry
. 458
3.17.1.1
Moyal Product and
Noncommutative
Algebra
. 458
3.17.1.2
Noncommutative
Space-Time Manifolds
459
3.17.1.3
Symmetries and Diffeomorphisms on
Deformed Spaces
. 462
3.17.1.4
Deformed Diffeomorphisms
. 465
3.17.1.5
Noncommutative
Space-Time Geometry
467
3.17.1.6
Star-Products and Expanded
Einstein-Hubert
Action
. 470
3.17.2
Synthetic Differential Geometry
. 473
3.17.2.1
Distributions
. 474
3.17.2.2
Synthetic Calculus in Euclidean Spaces
476
3.17.2.3
Spheres and Balls as Distributions
. 478
3.17.2.4
Stokes Theorem for Unit Sphere
. 480
3.17.2.5
Time Derivatives of Expanding Spheres
481
3.17.2.6
The Wave Equation
. 482
4.
Applied Bundle Geometry
485
4.1
Intuition Behind a Fibre Bundle
. 485
4.2
Definition of a Fibre Bundle
. . 486
4.3
Vector and
Affine
Bundles
. 491
4.3.1
The Second Vector Bundle of the Manifold
M
. . 495
4.3.2
The Natural Vector Bundle
. 496
4.3.3
Vertical Tangent and Cotangent Bundles
. 498
4.3.3.1
Tangent and Cotangent Bundles
Revisited
. 498
4.3.4
Affine
Bundles
. 500
4.4
Application: Semi-Riemannian Geometrical Mechanics
501
4.4.1
Vector-Fields and Connections
. 501
4.4.2
Hamiltonian Structures on the Tangent Bundle
. 503
4.5
К
-Theory and Its Applications
. 508
4.5.1
Topological
iť-Theory
. 508
4.5.1.1
Bott
Periodicity Theorem
. 509
4.5.2
Algebraic
Я
-Theory.
510
4.5.3
Chern Classes and Chern Character
. 511
4.5.4
Atiyah's View on K— Theory
. 515
4.5.5
Atiyah-Singer Index Theorem
. 518
xxvi
Applied
Differential
Geometry: A Modern
Introduction
4.5.6
The Infinite-Order Case
. 520
4.5.7
Twisted K-Theory and the Verlinde Algebra
. . 523
4.5.8
Application:
/ť-Theory
in String Theory
. 526
4.5.8.1
Classification of Ramond-Ramond
Fluxes
. 526
4.5.8.2
Classification of D-Branes
. 528
4.6
Principal Bundles
. 529
4.7
Distributions and Foliations on Manifolds
. 533
4.8
Application: Nonholonomic Mechanics
. 534
4.9
Application: Geometrical Nonlinear Control
. 537
4.9.1
Introduction to Geometrical Nonlinear Control
. . 537
419.2
Feedback Linearization
. 539
4.9.3
Nonlinear Controllability
. 547
4.9.4
Geometrical Control of Mechanical Systems
. . . 554
4.9.4.1
Abstract Control System
. 554
4.9.4.2
Global Controllability of Linear Control
Systems
. 555
4.9.4.3
Local Controllability of
Affine
Control
Systems
. 555
4.9.4.4
Lagrangian Control Systems
. 556
4.9.4.5
Lie-Adaptive Control
. 566
4.9.5
Hamiltonian Optimal Control and Maximum
Principle
. 567
4.9.5.1
Hamiltonian Control Systems
. 567
4.9.5.2
Pontryagin's Maximum Principle
. 570
4.9.5.3 Affine
Control Systems
. 571
4.9.6
Brain-Like Control Functor in
Biodynamics
. . . 573
4.9.6.1
Functor Control Machine
. 574
4.9.6.2
Spinal Control Level
. 576
4.9.6.3
Cerebellar Control Level
. 581
4.9.6.4
Cortical Control Level
. 584
4.9.6.5
Open Liouville
Neurodynamics
and
Biodynamical Self-Similarity
. 587
4.9.7
Brain-Mind Functorial Machines
. 594
4.9.7.1
Neurodynamical
2—
Functor
. 594
4.9.7.2
Solitary 'Thought Nets' and
'Emerging Mind'
. 597
4.9.8
Geometrodynamics of Human Crowd
. 602
4.9.8.1
Crowd Hypothesis
. 603
Contents xxvii
4.9.8.2 Geometrodynamics
of Individual
Agents
. 603
4.9.8.3
Collective Crowd Geometrodynamics
. . 605
4.10
M
ulti
vector-Fields and Tangent- Valued Forms
. 606
4.11
Application: Geometrical Quantization
. 614
4.11.1
Quantization of Hamiltonian Mechanics
. 614
4.11.2
Quantization of Relativistic Hamiltonian
Mechanics
. 617
4.12
Symplectic Structures on Fiber Bundles
. 624
4.12.1
Hamiltonian Bundles
. 625
4.12.1.1
Characterizing Hamiltonian Bundles
. . 625
4.12.1.2
Hamiltonian Structures
. 626
4.12.1.3
Marked Hamiltonian Structures
. 630
4.12.1.4
Stability
. 632
4.12.1.5
Cohomological Splitting
. 632
4.12.1.6
Homological Action of Ham(M) on
M
634
4.12.1.7
General Symplectic Bundles
. 636
4.12.1.8
Existence of Hamiltonian Structures
. . 637
4.12.1.9
Classification of Hamiltonian Structures
642
4.12.2
Properties of General Hamiltonian Bundles
. . . . 645
4.12.2.1
Stability
. 645
4.12.2.2
Functorial Properties
. 648
4.12.2.3
Splitting of Rational Cohomology
. 650
4.12.2.4
Hamiltonian Bundles and Gromov-
Witten Invariants
. 654
4.12.2.5
Homotopy Reasons for Splitting
. 659
4.12.2.6
Action of the Homology of (M) on
#»(M)
. 661
4.12.2.7
Cohomology of General Symplectic
Bundles
. 664
4.13
Clifford Algebras, Spinors and Penrose Twistors
. 666
4.13.1
Clifford Algebras and Modules
. 666
4.13.1.1
The Exterior Algebra
. 669
4.13.1.2
The Spin Group
. 672
4.13.1.3
4D Case
. . 672
4.13.2
Spinors
. 675
4.13.2.1
Basic Properties
. 675
4.13.2.2
4D Case
. 677
4.13.2.3
(Anti)
Self Duality
. 681
xxviii
Applied
Differential
Geometry: A Modern
Introduction
4.13.2.4
Herrai
tian
Structure on the
S
pinors
. . 686
4.13.2.5
Symplectic Structure on the Spinors
. . 689
4.13.3
Penrose Twistor Calculus
. 691
4.13.3.1
Penrose Index Formalism
. 691
4.13.3.2
Twistor Calculus
. 698
4.13.4
Application: Rovelli's Loop Quantum Gravity
. 701
4.13.4.1
Introduction to Loop Quantum Gravity
701
4.13.4.2
Formalism of Loop Quantum Gravity
. 708
4.13.4.3
Loop Algebra
. 709
4.13.4.4
Loop Quantum Gravity
. 711
4.13.4.5
Loop States and Spin Network States
. 712
4.13.4.6
Diagrammatic Representation of the
States
. 715
4.13.4.7
Quantum Operators
. 716
4.13.4.8
Loop v.s. Connection Representation
. 717
4.14
Application: Seiberg-
Witten
Monopole
Field Theory
. 718
4.14.1
SUSY Formalism
. 721
4.14.1.1 N = 2
Supersymmetry
. 721
4.14.1.2
ЛГ
= 2
Super-Action
. 721
4.14.1.3
Spontaneous Symmetry-Breaking
. . . 723
4.14.1.4
Holomorphy and Duality
. 724
4.14.1.5
The SW Prepotential
. 724
4.14.2
Clifford Actions, Dirac Operators and Spinor
Bundles
. 725
4.14.2.1
Clifford Algebras and Dirac Operators
. 727
4.14.2.2
Spin and
Spiile
Structures
. 729
4.14.2.3
Spinor Bundles
. 730
4.14.2.4
The Gauge Group and Its Equations
. . 731
4.14.3
Original SW Low Energy Effective Field Action
. 732
4.14.4
QED With Matter
. 735
4.14.5
QCD With Matter
. 737
4.14.6
Duality
. 738
4.14.6.1
Witten's Formalism
. 740
4.14.7
Structure of the Moduli Space
. 746
4.14.7.1
Singularity at Infinity
. 746
4.14.7.2
Singularities at Strong Coupling
. 747
4.14.7.3
Effects of a Massless
Monopole
. 748
4.14.7.4
The Third Singularity
. 749
Contents
XXIX
4.14.7.5 Monopole
Condensation
and
Confinement
. 750
4.14.8
Masses and Periods
. 752
4.14.9
Residues
. 754
4.14.10
SW
Monopole
Equations and Donaldson Theory
. 757
4.14.10.1
Topologica!
Invariance.
760
4.14.10.2
Vanishing Theorems
. 762
4.14.10.3
Computation on
Kahler
Manifolds
. . . 765
4.14.11
SW Theory and
Integrable
Systems
. 769
4.14.11.1
SU(N) Elliptic CM System
. 771
4.14.11.2
CM Systems Defined by Lie Algebras
. 772
4.14.11.3
Twisted CM-Systems Defined by
Lie Algebras
. 773
4.14.11.4
Scaling Limits of CM-Systems
. 774
4.14.11.5
Lax Pairs for CM-Systems
. 776
4.14.11.6
CM and SW Theory for SU(N)
. 779
4.14.11.7
CM and SW Theory for General
Lie Algebra
. 782
4.14.12
SW Theory and WDW Equations
. 784
4.14.12.1
WDW Equations
. 784
4.14.12.2
Perturbative SW Prepotentials
. 787
4.14.12.3
Associativity Conditions
. 789
4.14.12.4
SW Theories and
Integrable
Systems
. . 790
4.14.12.5
WDW Equations in SW Theories
. 793
5.
Applied Jet Geometry
797
5.1
Intuition Behind a Jet Space
. 797
5.2
Definition of a 1-Jet Space
. 801
5.3
Connections as Jet Fields
. 806
5.3.1
Principal Connections
. 815
5.4
Definition of a 2-Jet Space
. 818
5.5
Higher-Order Jet Spaces
. 822
5.6
Application: Jets and
Non-
Autonomous Dynamics
. . . 824
5.6.1
Geodesies
. 830
5.6.2
Quadratic Dynamical Equations
. 831
5.6.3
Equation of Free-Motion
. 832
5.6.4
Quadratic Lagrangian and Newtonian Systems
. . 833
5.6.5
Jacobi Fields
. 835
5.6.6
Constraints
. 836
xxx Applied Differential
Geometry: A Modem
Introduction
5.6.7
Time-Dependent Lagrangian Dynamics
. 841
516.8
Time-Dependent Hamiltonian Dynamics
. 843
5.6.9
Time-Dependent Constraints
. 848
5.6.10
Lagrangian Constraints
. 849
5.6.11
Quadratic Degenerate Lagrangian Systems
. 852
5.6.12
Time—Dependent
Integrable
Hamiltonian Systems
855
5.6.13
Time-Dependent Action-Angle Coordinates
. . . 858
5.6.14
Lyapunov Stability
. 860
5.6.15
First-Order Dynamical Equations
. 861
5.6.16
Lyapunov Tensor and Stability
. 863
5.6.16.1
Lyapunov Tensor
. 863
5.6.16.2
Lyapunov Stability
. 864
5.7
Application: Jets and Multi-Time Rheonomic
Dynamics
. 868
5.7.1
Relativistic Rheonomic Lagrangian Spaces
. 870
5.7.2
Canonical Nonlinear Connections
. 871
5.7.3
Cartan's Canonical Connections
. 874
5.7.4
General Nonlinear Connections
. 876
5.8
Jets and Action Principles
. 877
5.9
Application: Jets and Lagrangian Field Theory
. 883
5.9.1
Lagrangian Conservation Laws
. 888
5.9.2
General Covariance Condition
. 893
5.10
Application: Jets and Hamiltonian Field Theory
. 897
5.10.1
Covariant Hamiltonian Field Systems
. 899
5.10.2
Associated Lagrangian and Hamiltonian Systems
902
5.10.3
Evolution Operator
. 904
5.10.4
Quadratic Degenerate Systems
. 910
5.11
Application: Gauge Fields on Principal Connections
. . 913
5.11.1
Connection Strength
. 913
5.11.2
Associated Bundles
. 914
5.11.3
Classical Gauge Fields
. 915
5.11.4
Gauge Transformations
. 917
5.11.5
Lagrangian Gauge Theory
. 919
5.11.6
Hamiltonian Gauge Theory
. 920
5.11.7
Gauge Conservation Laws
. 923
5.11.8
Topologica!
Gauge Theories
. 924
5.12
Application: Modern Geometrodynamics
. 928
5.12.1
Stress-Energy-Momentum Tensors
. 928
5.12.2
Gauge Systems of Gravity and Fermion Fields
. . 955
Contents xxxi
5.12.3 Hawking-Penrose Quantum
Gravity and
Black
Holes
. 963
6.
Geometrical Path
Integrals
and Their Applications
983
6.1
Intuition Behind a Path Integral
. 984
6.1.1
Classical Probability Concept
. 984
6.1.2
Discrete Random Variable
. 984
6.1.3
Continuous Random Variable
. 984
6.1.4
General Markov Stochastic Dynamics
. 985
6.1.5
Quantum Probability Concept
. 989
6.1.6
Quantum Coherent States
. 991
6.1.7
Dirac's <bra\ket> Transition Amplitude
. . 992
6.1.8
Feynman's Sum-over-Histories
. 994
6.1.9
The Basic Form of a Path Integral
. 996
6.1.10
Application: Adaptive Path Integral
. 997
6.2
Path Integral History
. 998
6.2.1
Extract from Feynman's Nobel Lecture
. 998
6.2.2
Lagrangian Path Integral
. 1002
6.2.3
Hamiltonian Path Integral
. 1003
6.2.4
Feynman-Kac Formula
. 1004
6.2.5
Ito
Formula
. 1006
6.3
Standard Path-Integral Quantization
. 1006
6.3.1
Canonical versus Path-Integral Quantization
. . . 1006
6.3.2
Application: Particles, Sources, Fields and
Gauges
. 1011
6.3.2.1
Particles
.' 1011
6.3.2.2
Sources
. 1012
6.3.2.3
Fields
. 1013
6.3.2.4
Gauges
. 1013
6.3.3
Riemannian-Symplectic Geometries
. 1014
6.3.4
Euclidean Stochastic Path Integral
. 1016
6.3.5
Application: Stochastic Optimal Control
. 1020
6.3.5.1
Path-Integral Formalism
. 1021
6.3.5.2
Monte Carlo Sampling
. 1023
6.3.6
Application: Nonlinear Dynamics of Option
Pricing
. 1025
6.3.6.1
Theory and Simulations of Option
Pricing
. 1025
6.3.6.2
Option Pricing via Path Integrals
. . . 1029
xxxii
Applied
Differential
Geometry: A Modern
Introduction
б.З.б.З
Continuum Limit and American
Options
. 1035
6.3.7
Application: Nonlinear Dynamics of Complex
Nets
. 1036
6.3.7.1
Continuum Limit of the Kuramoto Net
1037
6.3.7.2
Path-Integral Approach to Complex
Nets
. 1038
6.3.8
Application: Dissipative Quantum Brain Model
1039
6.3.9
Application: Cerebellum as a Neural
Path-Integral
. 1043
6.3.9.1
Spinal Autogenetic Reflex Control
, . . 1045
6.3.9.2
Cerebellum
-
the Comparator
. 1047
6.3.9.3
Hamiltonian Action and Neural Path
Integral
. 1049
6.3.10
Path Integrals via Jets: Perturbative Quantum
Fields
. 1050
6.4
Sum over Geometries and Topologies
. 1055
6.4.1
Simplicial Quantum Geometry
. 1057
6.4.2
Discrete Gravitational Path Integrals
. 1059
6.4.3
Regge
Calculus
. 1061
6.4.4
Lorentzian Path Integral
. 1064
6.4.5
Application: Topological Phase Transitions
and Hamiltonian Chaos
. 1069
6.4.5.1
Phase Transitions in Hamiltonian
Systems
. 1069
6.4.5.2
Geometry of the Largest Lyapunov
Exponent
. 1072
6.4.5.3
Euler
Characteristics of Hamiltonian
Systems
. 1075
6.4.6
Application: Force-Field Psychodynamics
. . . 1079
6.4.6.1
Motivational Cognition in the Life
Space Foam
. 1080
6.5
Application: Witten's TQFT, SW-Monopoles and
Strings
. 1097
6.5.1
Topological Quantum Field Theory
. 1097
6.5.2
Seiberg-Witten Theory and TQFT
. 1103
6.5.2.1
S
W
Invariants and
Monopole
Equations
1103
6.5.2.2
Topological Lagrangian
. 1105
6.5.2.3
Quantum Field Theory
. 1107
Contents xxxiii
6.5.2.4
Dimensional
Reduction and
3D
Field
Theory
. 1112
6.5.2.5
Geometrical Interpretation
. 1115
6.5.3
TQFTs Associated with SW-Monopoles
. 1118
6.5.3.1
Dimensional Reduction
. 1122
6.5.3.2
TQFTs of
3D
Monopoles
. 1124
6.5.3.3
Non-Abelian Case
. 1135
6.5.4
Stringy Actions and Amplitudes
. 1138
6.5.4.1
Strings
. 1139
6.5.4.2
Interactions
. 1140
6.5.4.3
Loop Expansion
-
Topology of Closed
Surfaces
. 1141
6.5.5
Transition Amplitudes for Strings
. 1143
6.5.6
Weyl
Invariance
and Vertex Operator Formulation
1146
6.5.7
More General Actions
. 1146
6.5.8
Transition Amplitude for a Single Point Particle
. 1147
6.5.9
Witten's Open String Field Theory
. 1148
6.5.9.1
Operator Formulation of String Field
Theory
. 1149
6.5.9.2
Open Strings in Constant B—Field
Background
. 1151
6.5.9.3
Construction of Overlap Vertices
. 1154
6.5.9.4
Transformation of String Fields
. 1164
6.6
Application: Dynamics of Strings and Branes
. 1168
6.6.1
A Relativistic Particle
. 1169
6.6.2
A String
. 1171
6.6.3
A Brane
. 1173
6.6.4
String Dynamics
. 1175
6.6.5
Brane
Dynamics
. 1177
6.7
Application:
Topologica!
String Theory
. 1180
6.7.1
Quantum Geometry Framework
. 1180
6.7.2
Green-Schwarz Bosonic Strings and Branes
. 1181
6.7.3
Calabi-Yau Manifolds, Orbifolds and Mirror
Symmetry
. 1186
6.7.4
More on Topological Field Theories
. 1189
6.7.5
Topological Strings
. 1204
6.7.6
Geometrical Transitions
. 1221
6.7.7
Topological Strings and Black Hole Attractors
. . 1225
xxxiv
Applied Differential Geometry: A Modern Introduction
6.8
APPLICATION: Advanced Geometry and Topology of
String Theory
. 1232
6.8.1
String Theory and
Noncommutative
Geometry
. . 1232
6.8.1.1
Noncommutative
Gauge Theory
. 1233
6.8.1.2
Open Strings in the Presence of
Constant B-Field
. 1235
6.8.2
K-Theory Classification of Strings
. 1241
Bibliography
1253
Index
1295 |
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| author | Ivancevic, Vladimir G. Ivancevic, Tijana T. |
| author_facet | Ivancevic, Vladimir G. Ivancevic, Tijana T. |
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| building | Verbundindex |
| bvnumber | BV022958900 |
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| discipline | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-02T19:03:54Z |
| indexdate | 2024-07-09T21:08:35Z |
| institution | BVB |
| isbn | 9789812706140 9812706143 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016163291 |
| oclc_num | 441761570 |
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| physical | XXXIV, 1311 S. Ill., graph. Darst. |
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| spelling | Ivancevic, Vladimir G. Verfasser aut Applied differential geometry a modern introduction Vladimir G. Ivancevic ; Tijana T. Ivancevic New Jersey [u.a.] World Scientific 2007 XXXIV, 1311 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 1253 - 1293 Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Ivancevic, Tijana T. Verfasser aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163291&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ivancevic, Vladimir G. Ivancevic, Tijana T. Applied differential geometry a modern introduction Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Applied differential geometry a modern introduction |
| title_auth | Applied differential geometry a modern introduction |
| title_exact_search | Applied differential geometry a modern introduction |
| title_exact_search_txtP | Applied differential geometry a modern introduction |
| title_full | Applied differential geometry a modern introduction Vladimir G. Ivancevic ; Tijana T. Ivancevic |
| title_fullStr | Applied differential geometry a modern introduction Vladimir G. Ivancevic ; Tijana T. Ivancevic |
| title_full_unstemmed | Applied differential geometry a modern introduction Vladimir G. Ivancevic ; Tijana T. Ivancevic |
| title_short | Applied differential geometry |
| title_sort | applied differential geometry a modern introduction |
| title_sub | a modern introduction |
| topic | Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Geometry, Differential Textbooks Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163291&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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