Verfügbarkeit: Analysis, manifolds and physics | Technische Universität München

Analysis, manifolds and physics: 1 Basics

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Bibliographische Detailangaben
Beteiligte Personen:	Choquet-Bruhat, Yvonne 1923-2025 (VerfasserIn), DeWitt-Morette, Cécile 1922-2017 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Amsterdam [u.a.] North-Holland 1996
Ausgabe:	Rev. ed., reprint. with minor corr.
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007481900&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XX, 630 S. graph. Darst.
ISBN:	0444860177 9780444860170

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Datensatz im Suchindex

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adam_text	CONTENTS I, Review oř Fundamental Notions of Analysis 1 A. Set Theory, Definitions 1 1. Sets 1 2. Mappings 2 3. Relations 5 4. Orderings 5 B. Aigebraic Structures, Definitions 6 1. Groups 7 2. Rings 8 3. Modules 8 4. Algebras 9 5. Linear spaces 9 С Topology 11 1. Definitions 11 2. Separation 13 3. Base 14 4. Convergence 14 5. Covering and compactness 15 6. Connectedness 16 7. Continuous mappings 17 8. Multiple connectedness 19 9. Associated topologies 20 10. Topology related to other structures 21 11. Metric spaces 23 metric spaces 23 Cauchy sequence; completeness 25 12. Banach spaces 26 normed vector spaces 27 Banach spaces 28 strong and weak topology; compactedness 29 13. Hubert spaces 30 Đ. Integration 31 1. Introduction 32 2. Measures 33 3. Measure spaces 34 ix CONTENTS 4. Measurable functions 40 5. Integrable functions 41 6. Integration on locally compact spaces 46 7. Signed and complex measures 49 8. Integration of vector valued functions 50 9. V space 52 10. L" space 53 E. Key Theorems in Linear Functional Analysis 57 1. Bounded linear operators 57 2. Compact operators 61 3. Open mapping and closed graph theorems 63 Problems and Exercises 64 Problem 1: Clifford algebra; Spin(4) 64 Exercise 2: Product topology 68 Problem 3: Strong and weak topologies in L1 69 Exercise 4: Holder spaces 70 See Problem VI 4: Application to the Schrödinger equation II. Differential Calculus on Banach Spaces 71 A. Foundations 71 1. Definitions. Taylor expansion 71 2. Theorems 73 3. Diffeomorphisms 74 4. The Euler equation 76 5. The mean value theorem 78 6. Higher order differentials 79 B. Calculus of Variations 82 1. Necessary conditions for minima 82 2. Sufficient conditions 83 3. Lagrangian problems 86 С Implicit Function Theorem. Inverse Function Theorem 88 1. Contracting mapping theorems 88 2. Inverse function theorem 90 3. Implicit function theorem 91 4. Global theorems 92 D. Differential Equations 94 1. First order differential equation 94 2. Existence and uniqueness theorems for the lipschitzian case 95 CONTENTS xi Problems and Exercises 98 Problem 1: Banach spaces, first variation, linearized equation 98 Problem 2: Taylor expansion of the action; Jacobi fields; the Feynman-Green function; the Van Vleck matrix; conjugate points; caustics 100 Problem 3: Euler-Lagrange equation; the small disturbance equation; the soap bubble problem; Jacobi fields 105 III. Differentiable Manifolds, Finite Dimensional Case 111 A. Definitions 111 1. Differentiable manifolds 111 2. Diffeomorphisms 115 3. Lie groups 116 B. Vector Fields; Tensor Fields 117 1. Tangent vector space at a point 117 tangent vector as a derivation 118 tangent vector defined by transformation properties 120 tangent vector as an equivalence class of curves 121 images under differentiable mappings 121 2. Fibre bundles 124 definition 125 bundle morphisms 127 tangent bundle 127 frame bundle 128 principal fibre bundle 129 3. Vector fields 132 vector fields 132 moving frames 134 images under diffeomorphisms 134 4. Covariant vectors; cotangent bundles 135 dual of the tangent space 135 space of differentials 137 cotangent bundle 138 reciprocal images 138 5. Tensors at a point 138 tensors at a point 138 tensor algebra 140 6. Tensor bundles; tensor fields 142 С Groups of Transformations ' 143 1. Vector fields as generators of transformation groups 143 2. Lie derivatives 147 3. Invariant tensor fields 150 xii CONTENTS D. Lie Groups 152 1. Definitions; notations 152 2. Left and right translations; Lie algebra; structure constants 155 3. One-parameter subgroups 158 4. Exponential mapping; Taylor expansion; canonical coordinates 160 5. Lie groups of transformations; realization 162 6. Adjoint representation 166 7. Canonical form, Maurer-Cartan form 168 Problems and Exercises 169 Problem 1: Change of coordinates on a fiber bundle, configuration space, phase space 169 Problem 2: Lie algebras of Lie groups 172 Problem 3: The strain tensor 177 Problem 4: Exponential map; Taylor expansion; adjoint map; loft and right differentials; Haar measure 178 Problem 5: The group manifolds of SO(3) and SU(2) 181 Problem 6: The 2-sphere 190 IV. Integration on Manifolds 195 A. Exterior Differential Forms 195 1. Exterior algebra 195 exterior product 196 local coordinates; strict components 197 change of basis 199 2. Exterior differentiation 200 3. Reciprocal image of a form (pull back) 203 4. Derivations and antiderivations 205 definitions 206 interior product 207 5. Forms defined on a Lie group 208 invariant forms 208 Maurer-Cartan structure equations 208 6. Vector valued differential forms 210 B. Integration 212 1. Integration 212 orientation 212 odd forms 212 integration of и -forms in R" 213 partitions of unity 214 properties of integrals 215 2. Stokes' theorem 216 p-chains 217 integrals of p-forms on p-chains 217 boundaries 218 CONTENTS mappings of chains 219 proof of Stokes' theorem 221 3. Global properties 222 homology and cohomology 222 О -forms and 0-chains 223 Betti numbers 224 Poincaré lemmas 224 de Rham and Poincaré duality theorems 226 С Exterior Differential Systems 229 1. Exterior equations 229 2. Single exterior equation 229 3. Systems of exterior equations 232 ideal generated by a system of exterior equations 232 algebraic equivalence 232 solutions 233 examples 235 4. Exterior differential equations 236 integral manifolds 236 associated Pfaff systems 237 generic points 238 closure 238 5. Mappings of manifolds 239 introduction 239 immersion 241 embedding 241 submersion 242 6. Pfaff systems 242 complete integrability 243 Frobenius theorem 243 integrability criterion 245 examples 246 dual form of the Frobenius theorem 248 7. Characteristic system 250 characteristic manifold 250 example: first order partial differential equations 250 complete integrability 253 construction of integral manifolds 254 Cauchy problem 256 examples 259 8. Invariants 261 invariant with respect to a Pfaff system 261 integral invariants 263 9. Example: Integral invariants of classical dynamics 265 Liouville theorem 266 canonical transformations 267 xiv CONTENTS 10. Symplectic structures and hamiltonian systems 267 Problems and Exercises 270 Problem 1: Compound matrices 270 Problem 2: Poincaré lemma, Maxwell equations, wormholes 271 Problem 3: Integral manifolds 271 Problem 4: First order partial differential equations, Hamilton-Jacobi equations, lagrangian manifolds 272 Problem 5: First order partial differential equations, catastrophes 277 Problem 6: Darboux theorem 281 Problem 7: Time dependent hamiltonians 283 See Problem VI 11 paragraph c: Electromagnetic shock waves V. Riemannian Manifolds. Kählerian Manifolds 285 A. The Riemannian Structure 285 1. Preliminaries 285 metric tensor 285 hyperbolic manifold 287 2. Geometry of submanifolds, induced metric 290 3. Existence of a riemannian structure 292 proper structure 292 hyperbolic structure 293 Euler-Poincaré characteristic 293 4. Volume element. The star operator 294 volume element 294 star operator 295 5. Isometries 298 B. Linear Connections 300 1. Linear connections 300 covariant derivative 301 connection forms 301 parallel translation 302 affine geodesic 302 torsion and curvature 305 2. Riemannian connection 308 definitions 309 locally flat manifolds 310 3. Second fundamental form 312 4. Differential operators 316 exterior derivative 316 operator б 317 divergence 317 laplacian 318 CONTENTS xv C. Geodesies 320 1. Arc length 320 2. Variations 321 Euler equations 323 energy integral 324 3. Exponential mapping 325 definition 325 normal coordinates 326 4. Geodesies on a proper riemannian manifold 327 properties 327 geodesic completeness 330 5. Geodesies on a hyperbolic manifold 330 D. Almost Complex and Kăhlerian Manifolds 330 Problems and Exercises 336 Problem 1: Maxwell equation; gravitational radiation 336 Problem 2: The Schwarzschild solution 341 Problem 3: Geodetic motion; equation of geodetic deviation; exponentiation; conjugate points 344 Problem 4: Causal structures; conformai spaces; Weyl tensor 350 Vbis. Connections on a Principal Fibre Bundle A. Connections on a Principal Fibre Bundle 357 1. Definitions 357 2. Local connection 1-forms on the base manifold 362 existence theorems 362 section canonically associated with a trivialization 363 potentials 364 change of trivialization 364 examples 366 3. Covariant derivative 367 associated bundles 367 parallel transport 369 covariant derivative 370 examples 371 4. Curvature 372 definitions 372 Cartan structural equation 373 local curvature on the base manifold 374 field strength 375 Bianchi identities 375 5. Linear connections 376 definition 376 soldering form, torsion form 376 xvi CONTENTS torsion structural equation 376 standard horizontal (basic) vector field 378 curvature and torsion on the base manifold 378 bundle homomorphism 380 metric connection 381 B. Holonomy 381 1. Reduction 381 2. Holonomy groups 386 C. Characteristic Classes and Invariant Curvature Integrals 390 1. Characteristic classes 390 2. Gauss-Bonnet theorem and Chern numbers 395 3. The Atiyah-Singer index theorem 396 Problems and Exercises 401 Problem 1: The geometry of gauge fields 401 Problem 2: Charge quantization. Monopoles 408 Problem 3: Instanton solution of euclidean SU(2) Yang-Mills theory (connection on a non-trivial SU(2) bundle over S4) 411 Problem 4: Spin structure; spinors; spin connections ^15 VI. Distributions 423 A. Test Functions 423 1. Seminorms 423 definitions 423 Hahn-Banach theorem 424 topology defined by a family of seminorms 424 2. Si-spaces 427 definitions 427 inductive limit topology 429 convergence in 3)m(U) and B(U) 430 examples of functions in 2) 431 truncating sequences 434 density theorem 434 B. Distributions 435 1. Definitions 435 distributions 435 measures; Dirac measures and Leray forms 437 distribution of order ρ 439 support of a distribution 441 distributions with compact support 441 CONTENTS xvii 2. Operations on distributions 444 sum 444 product by C' function 444 direct product 445 derivations 446 examples 447 inverse derivative 450 3. Topology on Э' 453 weak star topology 453 criterion of convergence 454 4. Change of variables in R" 456 change of variables in R" 456 transformation of a distribution under a diffeomorphism 457 invariance 459 5. Convolution 459 convolution algebra L'OR") 459 convolution algebra 2)'+ and @'~ 462 derivation and translation of a convolution product 464 regularization 465 support of a convolution 465 equations of convolution 466 differential equation with constant coefficients 469 systems of convolution equations 470 kernels 471 6. Fourier transform 474 Fourier transform of integrable functions 474 tempered distributions 476 Fourier transform of tempered distributions 476 Paley-Wiener theorem 477 Fourier transform of a convolution 478 7. Distribution on a C' paracompact manifold 480 8. Tensor distributions 482 С Sobolev Spaces and Partial Differential Equations 486 1. Sobolev spaces 486 properties 487 density theorems 488 W/ spaces 489 Fourier transform 490 Plancherel theorem 490 Sobolev's inequalities 491 2. Partial differential equations 492 definitions 492 Cauchy-Kovalevski theorem 493 classifications 494 3. Elliptic equations; laplacians 495 CONTENTS elementary solution of Laplace's equation 495 subharmonic distributions 496 potentials 496 energy integral, Green's formula, unicity theorem 499 Liouville's theorem 500 boundary-value problems 502 Green function 503 introduction to hilbertian methods; generalized Dirichlet problem 505 hilbertian methods 507 example: Neumann problem 509 4. Parabolic equations 510 heat diffusion 510 5. Hyperbolic equation; wave equations 511 elementary solution of the wave equation 511 Cauchy problem 512 energy integral, unicity theorem 513 existence theorem 515 6. Leray theory of hyperbolic systems 516 7. Second order systems; propagators 522 Problems and Exercises 525 Problem 1: Bounded distributions 525 Problem 2: Laplacian of a discontinuous function 527 Exercise 3: Regularized functions 528 Problem 4: Application to the Schrödinger equation 528 Exercise 5: Convolution and linear continuous responses 530 Problem 6: Fourier transforms of exp(-x2) and exp (ix2) 531 Problem 7: Fourier transforms of Heaviside functions and Pv(l/x) 532 Problem 8: Dirac bitensors 533 Problem 9: Legendre condition 533 Problem 10: Hyperbolic equations; characteristics 534 Problem 11: Electromagnetic shock waves 535 Problem 12: Elementary solution of the wave equation 538 Problem 13: Elementary kernels of the harmonic oscillator 538 VII. Differentiable Manifolds, Infinite Dimensional Case 543 A. Infinite-Dimensional Manifolds 543 1. Definitions and general properties 543 E-manifolds 543 diflerentiable functions 544 tangent vector 544 vector and tensor field 545 differential of a mapping 546 CONTENTS xix submanifold 547 immersion, embedding, submersion 549 flow of a vector field 551 differential forms 551 2. Symplectic structures and hamiltonian systems 552 definitions 552 complex structures 552 canonical symplectic form 554 symplectic transformation 554 hamiltonian vector field 554 conservation of energy theorem 555 riemannian manifolds 555 B. Theory of Degree; Leray-Schauder Theory 556 1. Definition for finite dimensional manifolds 557 degree 557 integral formula for the degree of a function 558 continuous mappings 560 2. Properties and applications 561 fundamental theorem 561 Borsuk's theorem 562 Brouwer's fixed point theorem 562 product theorem 563 3. Leray-Schauder theory 563 definitions 563 compact mappings 564 degree of a compact mapping 564 Schauder fixed point theorem 565 Leray-Schauder theorem 565 С Morse Theory 567 1. Introduction 567 2. Definitions and theorems 567 3. Index of a critical point 571 4. Critical neck theorem 572 D. Cylindrical Measures, Wiener Integral 573 1. Introduction 573 2. Promeasures and measures on a locally convex space 575 projective system 575 promeasures 576 image of a promeasure 578 integration with respect to a promeasure of a cylindrical function 578 Fourier transforms 579 3. Gaussian promeasures 581 xx CONTENTS gaussian measures on R" 581 gaussian promeasures 582 gaussian promeasures on Hubert spaces 583 4. The Wiener measure 583 Wiener integral 586 sequential Wiener integral 587 Problems and Exercises 589 Problem A: The Klein-Gordon equation 589 Problem B: Application of the Leray-Schauder theorem 591 Problem Cl: The Reeb theorem 592 Problem C2: The method of stationary phase 593 Problem DI: A metric on the space of paths with fixed end points 596 Problem D2: Measures invariant under translation 597 Problem D3: Cylindrical σ -field of Cfta, b]) 597 Problem D4: Generalized Wiener integral of a cylindrical function 598 References 603 Symbols 611 Index 617 CONTENTS TO PART Π: 92 APPLICATIONS Preface ν Contents vii Conventions xi I. REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS 1 1. Graded algebras 1 2. Berezinian 3 3. Tensor product of algebras 5 4. Clifford algebras 6 5. Clifford algebra as a coset of the tensor algebra . 14 6. Fierz identity 15 7. Pin and Spin groups 17 8. Weyl spinors, helicity operator; Majorána pinors, charge conjugation 27 9. Representations of Spin(n, m), n + m odd 33 10. Dirac adjoint 36 11. Lie algebra of Pin(n, m) and Spin(n, m) 37 12. Compact spaces 39 13! Compactness in weak star topology 40 14. Homotopy groups, general properties 42 15. Homotopy of topological groups 46 16. Spectrum of closed and self-adjoint linear operators 47 II. DIFFERENTIAL CALCULUS ON BANACH SPACES 51 1. Supersmooth mappings 51 2. Berezin integration; Gaussian integrals 57 3. Noether's theorems I 64 4. Noether's theorems II 71 5. Invariance of the equations of motion 79 CONTENTS 6. String action 82 7. Stress-energy tensor; energy with respect to a timelike vector field 83 III. DIFFERENTIABLE MANIFOLDS 91 1. Sheaves 91 2. Differentiable submanifolds 91 3. Subgroups of Lie groups. When are they Lie sub¬ groups? 92 4. Cartan-Killing form on the Lie algebra 's of a Lie group G 93 5. Direct and semidirect products of Lie groups and their Lie algebra 95 6. Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields 102 7. Homogeneous spaces; symmetric spaces 103 8. Examples of homogeneous spaces, Stiefel and Grass- mann manifolds 108 9. Abelian representations of nonabelian groups 110 10. Irreducibility and reducibility 111 11. Characters 114 12. Solvable Lie groups 114 13. Lie algebras of linear groups 115 14. Graded bundles 118 IV. INTEGRATION ON MANIFOLDS 127 1. Cohomology. Definitions and exercises 127 2. Obstruction to the construction of Spin and Pin bun¬ dles; Stiefel-Whitney classes 134 3. Inequivalent spin structures 150 4. Cohomology of groups 158 5. Lifting a group action 161 6. Short exact sequence; Weyl Heisenberg group 163 7. Cohomology of Lie algebras 167 8. Quasi-linear first-order partial differential equation 171 9. Exterior differential systems (contributed by B. Kent Harrison) 173 10. Bäcklund transformations for evolution equations (con¬ tributed by N.H. Ibragimov) 181 11. Poisson manifolds I 184 CONTENTS 12. Poisson manifolds II (contributed by C. Moreno) 200 13. Completely integrable systems (contributed by С Moreno) 219 V. RIEMANNIAN MANIFOLDS. KÄHLERIAN MANIFOLDS 235 1. Necessary and sufficient conditions for Lorentzian sig¬ nature 235 2. First fundamental form (induced metric) 238 3. Kilting vector fields 239 4. Sphère S" 240 5. Curvature of Einstein cylinder 244 6. Conformai transformation of Yang-Mills, Dirac and Higgs operators in d dimensions 244 7. Conformai system for Einstein equations 249 8. Conformai transformation of nonlinear wave equations 256 9. Masses of "homothetic" space-time 262 10. Invariant geometries on the squashed seven spheres 263 11. Harmonic maps 274 12. Composition of maps 281 13. Kaluza-Klein theories 286 14. Kahler manifolds; Calabi-Yau spaces 294 V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE 303 1. An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base 303 2. Gauge transformations 305 3. Hopf fibering S3-*S2 307 4. Subbundles and reducible bundles 308 5. Broken symmetry and bundle reduction, Higgs mech¬ anism 310 6. The Euler-Poincaré characteristic 321 7. Equivalent bundles 334 8. Universal bundles. Bundle classification 335 9. Generalized Bianchi identity 340 10. Chem-Simons classes 340 11. Cocycles on the Lie algebra of a gauge group; Anomalies 349 CONTENTS 12. Virasoro representation of S£(Diñ S1). Ghosts. BRST operator 363 VI. DISTRIBUTIONS 373 1. Elementary solution of the wave equation in eř-dimen- sional spacetime 373 2. Sobolev embedding theorem 377 3. Multiplication properties of Sobolev spaces 386 4. The best possible constant for a Sobolev inequality on W,n>3 (contributed by H. Grosse) 389 5. Hardy-Littlewood-Sobolev inequality (contributed by H. Grosse) 391 6. Spaces Hs 8 (IR") 393 7. Spaces Hs (Г) and Hs „ (Un) 396 8. Completeness of a ball of Wps in Wf_j 398 9. Distribution with laplacian in L2(U") 399 10. Nonlinear wave equation in curved spacetime 400 11. Harmonic coordinates in general relativity 405 12. Leray theory of hyperbolic systems. Temporal gauge in general relativity 407 13. Einstein equations with sources as a hyperbolic system 413 14. Distributions and analyticity: Wightman distributions and Schwinger functions (contributed by C. Doering) 414 15. Bounds on the number of bound states of the Schröd- inger operator 425 16. Sobolev spaces on riemannian manifolds 428 Subject Index 433 Errata to Part I 439
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spelling	Choquet-Bruhat, Yvonne 1923-2025 Verfasser (DE-588)128916206 aut Analysis, manifolds and physics 1 Basics by Yvonne Choquet-Bruhat, Cécile DeWitt-Morette Rev. ed., reprint. with minor corr. Amsterdam [u.a.] North-Holland 1996 XX, 630 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier DeWitt-Morette, Cécile 1922-2017 Verfasser (DE-588)141948361 aut (DE-604)BV004271058 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007481900&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Choquet-Bruhat, Yvonne 1923-2025 DeWitt-Morette, Cécile 1922-2017 Analysis, manifolds and physics
title	Analysis, manifolds and physics
title_auth	Analysis, manifolds and physics
title_exact_search	Analysis, manifolds and physics
title_full	Analysis, manifolds and physics 1 Basics by Yvonne Choquet-Bruhat, Cécile DeWitt-Morette
title_fullStr	Analysis, manifolds and physics 1 Basics by Yvonne Choquet-Bruhat, Cécile DeWitt-Morette
title_full_unstemmed	Analysis, manifolds and physics 1 Basics by Yvonne Choquet-Bruhat, Cécile DeWitt-Morette
title_short	Analysis, manifolds and physics
title_sort	analysis manifolds and physics basics
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