The Riemann legacy: Riemannian ideas in mathematics and physics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1997
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| Series: | Mathematics and its applications
417 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Poln. übers. |
| Physical Description: | XXII, 717 S. graph. Darst. |
| ISBN: | 079234636X |
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| 100 | 1 | |a Maurin, Krzysztof |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The Riemann legacy |b Riemannian ideas in mathematics and physics |c by Krzysztof Maurin |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1997 | |
| 300 | |a XXII, 717 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 417 | |
| 500 | |a Aus dem Poln. übers. | ||
| 600 | 1 | 4 | |a Riemann, Bernhard <1826-1866> |
| 600 | 1 | 7 | |a Riemann, Bernhard |d 1826-1866 |0 (DE-588)118600869 |2 gnd |9 rswk-swf |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Geometry, Riemannian | |
| 650 | 4 | |a Mathematical physics | |
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Contents
Foreword: Riemann's Geometrie Ideas and their Role in
Mathematics and Physics xiii
I Riemannian Ideas in Mathematics and Physics 1
1 Gauss Inner Curvature of Surfaces 3
1.1 Parallel transport and linear (affine) connection 6
1.2 Vector bundles and operations on them 8
1.3 Riemann surfaces 11
1.4 Riemannian connection. Levi Civita connection 16
1.5 Geodesics in Riemann space (manifold) (M,g) as lines of ex
tremal length. Euler Lagrange equation 20
1.6 Jacobi fields (curvature and geodesics) 22
2 Sectional Curvature. Spaces of Constant Curvature. Weyl Hypothe
sis 24
2.1 Ovals 28
2.2 Riemannian manifolds as metric Spaces (Hopf Rinow).
Geodesic completeness 31
2.3 Symmetrie Spaces 33
2.4 Bounded regions in complex plane. Bergman metric (for the
first time) 36
2.5 Siegel half space and Siegel disc 38
2.6 Jacobi fields once again. Focal points 42
3 Cohomology of Riemann Spaces. Theorems of de Rham, Hodge, Ko
daira 48
3.1 Homology. Cohomology. De Rham cohomology 49
3.2 Hodge theory of harmonic forms 53
v
3.3 Hodge decomposition 55
3.4 The method of heat transport (diffusion equation) 57
3.5 The Euler Poincare characteristic (Euler number) 61
3.6 Index theorem (for the first time) 62
3.7 Sobolev Spaces. Theorems of Rellich, Sobolev, and Gärding . 64
3.8 Weitzenbröck formulas 67
3.9 Euler form. Hopf theorem on index of vector field 69
3.10 Poincare duality. Künneth theorem 70
3.11 Intersection number (Kronecker index) of two cycles 73
3.12 Index of vector a field and degree of mapping. Kronecker
integral 77
3.13 Relation between Morse index and index of a vector field . . 82
4 Chern Gauss Bonnet theorem 84
4.1 Allendorfer Weil formula 87
5 Curvature and Topology or Characteristic Forms of Chern, Pontria
gin, and Euler 90
5.1 Chern forms 90
5.2 Pontriagin forms. Pfaffian Rv. Chern theorem once again . . 97
5.3 Hirzenbruch signature theorem 99
5.4 General index theorem (Atiyah Singer) 100
6 Kahler Spaces. Bergman Metrics. Harish Chandra Cartan Theorem.
Siegel Space (once again!) 102
6.1 Calabi hypothesis and Calabi Yau Spaces 105
6.2 Bergman metrics on bounded domains 107
6.3 Imbedding in projective Spaces. Kodaira theorem 108
6.4 Homogeneous complex spaces and bounded domains 111
6.5 Symmetrie spaces 114
6.6 Spectral geometry 116
II General Structures of Mathematics 119
1 Differentiable Structures. Tangent Spaces. Vector Fields 121
2 Projective (Inverse) Limits of Topological Spaces 134
3 Inductive Limits. Presheaves. Covering Defined by Presheaf 137
4 Algebras. Groups, Tensors, Clifford, Grassmann, and Lie Algebras.
Theorems of Bott Milnor, Wedderburn, and Hurwitz. 148
5 Fields and their Extensions 163
6 Galois Theory. Solvable Groups 175
7 Ruler and Compass Constructions. Cyclotomic Fields. Kronecker
Weber Theorem 184
8 Algebraic and Transcendental Elements 189
9 Weyl principle 191
10 Topology of Compact Lie Groups 194
11 Representations of Compact Lie Groups 196
12 Nilpotent, Semimple, and Solvable Lie Algebras 210
13 Reflections, Roots, and Weights. Coxeter and Weyl groups 216
13.1 Weights of representations of Lie algebra 220
13.2 Classification of root Systems. Coxeter diagrams 220
13.3 Relation with semisimple complex Lie algebras 223
14 Covariant Differentiation. Parallel Transport. Connections 230
15 Remarks on Rieh Mathematical Structures of Simple Notions of
Physics Based on Example of Analytical Mechanics 237
16 Tangent Bündle TM. Vector, Fiber, Tensor and Tensor Densities,
and Associate Bundles 242
17 G' spaces. Group Representations 253
18 Principal and Associated Bundles 258
19 Induced Representations and Associated Bundles 265
20 Vector Bundles and Locally Free Sheaves 268
21 Axiom of Covering Homotopy 271
22 Serre Fibering. General Theory of Connection. Corollaries 274
23 Homology. Cohomology. de Rham Cohomology 281
24 Cohomology of Sheaves. Abstract de Rham Theorem 286
25 Homotopy Group 7rjt(A",xo). Hopf Fibering. Serre Theorem on Exact
Sequence of Homotopy Groups of a Fibering 292
26 Various Benefits of Characteristic Classes (Orientability, Spin Struc
tures). Clifford Groups, Spin Group 297
27 Divisors and Line Bundles. Algebraic and Abelian Varieties 303
28 General Abelian Varieties and Theta Function 310
28.1 Theta functions 313
28.2 Strictly transcendental extensions. Transcendental degree . . 316
29 Theorems on Algebraic Dependence 318
III The Idea of the Riemann Surface 325
29.1 Introduction 327
29.2 Fredholm Noether Operators. Parametrices 329
29.3 Proof of Riemann Roch theorem 333
29.4 The fundamental theorem for compact surfaces 341
29.5 Embedding of Riemann surfaces 343
29.6 Hyperelliptic surfaces. Hyperelliptic involutions 345
29.7 Weierstrass points. Wronskian 347
29.8 Hyperelliptic involution 349
29.9 Clifford theorem 351
29.10Riemann bilinear relations. Abel Jacobi map 351
29.11Linear bundles on complex tori: Appel Humbert theorem . . 355
29.120 functions. The great Riemann theorems: 'Abel theorem',
'Jacobi Inversion', and '\ theorem' 357
IV Riemann and Calculus of Variations 361
1 Introduction 363
1.1 General criteria for existence of minimizers of functionals . . 366
1.2 Convexity and weak lower semi continuity 368
2 The Plateau Problem 370
2.1 Coercity of Dirichlet integral 370
2.2 The Rado Douglas solution of Plateau problem 371
2.3 Riemann mapping theorem and Plateau problem 378
2.4 Representation formulas for minimal surfaces. Enneper
Weierstrass theorem. Scherk surface 380
2.5 Minimal surfaces and value distribution theory 384
2.6 Some properties of harmonic maps. Theorems of Eells
Sampson, Hartman, and corollaries 388
3 Teichmüller Theory. Riemann Moduli Problem 396
3.1 Teichmüller metric 398
3.2 The analytic structure of the Teichmüller space Tp 398
3.3 The moduli Space 399
4 Riemannian Approach to Teichmüller Theory. Harmonic Maps and
Teich müller Space 401
4.1 Hermitian hyperbolic geometry of Kobayashi 414
4.2 Hyperbolic complex analysis 418
4.3 Hyperbolicity of the Teichmüller space 418
4.4 Kobayashi pseudodistance. Kobayashi hyperbolic Spaces . . . 419
4.5 Invariant metrics of Teichmüller space 421
4.6 Harmonic Beltrami differentials on (M, g) 422
4.7 Wolpert formulas for Petersson Weil form 427
4.8 Generalization to higher dimensions 430
4.9 Metrics on Teichmüller space (general remarks) 432
4.10 The period map. Royden theorems 434
4.11 The period map and Torelli theorems 436
5 Teichmüller theory and Plateau Douglas problem 438
6 Rescuing Riemann's Dirichlet Principle. Potential Theory 445
6.1 Subharmonic functions. Riesz decomposition 446
6.2 Poisson integral and Harnack theorems 447
6.3 History of the potential theory 449
6.4 Perron method 451
6.5 Rado theorem. Theorem of Poincare Volterra 453
7 The Royal Road to Calculus of Variations (Constantin Caratheodory) 457
7.1 Introduction 457
7.2 Fields 459
7.3 An equivalent problem 461
7.4 Integrability conditions. Geodesic fields. (Independent)
Hilbert integral 462
7.5 Weierstrass excess function and condition for strong minimum 463
7.6 Legendre condition for weak minimum 463
7.7 Complete figure of variational problem 464
7.8 Problems with free endpoints. Broken extremals 466
7.9 Legendre transformation. Canonical equations of Hamilton.
Hilbert integral in canonical coordinates. Hamilton Jacobi
theory 468
7.10 Physical meaning of functions H, S, and L 470
7.11 Lagrange bracket and geodesic fields 472
7.12 Canonical transformations 473
7.13 Caustics. 'Enveloppensatz'of Caratheodory. Singularities . . 476
7.14 Finsler geometry and geometric optics 477
7.15 General Huygens principle and Finsler geometry 479
7.16 Field theories for calculus of Variation for multiple integrals . 482
7.17 Lepage theory of geodesic fields 485
7.18 Caratheodory and thermodynamics (second law). Pfaff prob¬
lem and Frobenius theorem 490
7.19 Caratheodory and the beginning of calculus of variations . . . 492
8 Symplectic and Contact Geometries. Conservation Laws 497
8.1 Introduction 497
8.2 Lie approach to hamiltonian mechanics 503
8.3 Conservation laws and 'Postulates of impotence' 505
8.4 Momentum map and symplectic reduction. (Reduction of
phase space for Systems with symmetries) 506
8.5 Hyperkähler quotients 511
9 Direct Methods in Calculus of Variations for Manifolds with Isome¬
tries. Equivariant Sobolev Theorems. Yamabe Problem and its Re¬
lation to General Relativity 513
V Riemann and Complex Geometry 523
1 Introduction 525
2 On Complex Analysis in Several Variables 528
3 Ellipticity, Runge Property, and Runge Type Theorems 543
4 Hörmander Method in Complex Analysis 552
5 Wirtinger Theorems. Metric Theory of Analytic Sets 560
6 The Problem of Poincare and the Cousin Problems 567
7 Ringed Spaces and General Complex Spaces 578
8 Construction of Complex Spaces by Gluing and by Taking Quotient 596
8.1 Construction of complex Spaces by gluing 598
8.2 On deformations of regulär families of complex structures
(Grauert theory) 599
8.3 Grauert solution of main problems of deformation theory of
complex structures 605
8.4 On differential calculus on complex Spaces 606
8.5 From Riemann period relations to theorems of Kodaira and
Grauert 609
8.6 Concluding remarks 613
9 Differential Geometry of Holomorphic Vector Bundles over Compact
Riemann Surfaces and Kahler manifolds. Stable Vector Bundles,
Hermite Einstein Connections, and their Moduli Spaces 615
9.1 Fiat bundles and flat connections 619
9.2 Moduli Spaces of H E structures 624
9.3 Hermite Einstein metrics (structures) as critical points of
Donaldson functional (variational theory of H E connections) 627
9.4 Kahler structures on moduli space MH~E{E) 635
VI Riemann and Number Theory 649
1 Introduction 651
1.1 Introduction 651
1.2 Automorphic forms, modular functions 653
2 The Riemann £ function 655
2.1 L functions of cusp forms 657
3 Hecke Theory 659
3.1 Petersson Scalar Product 659
3.2 Hecke Operators 660
3.3 Hecke L series 664
3.4 Ramanujan Petersson conjecture and Deligne theorein . 667
3.5 Hecke theory for congruence subgroups 668
3.6 Congruence subgroups F C F(l), their modular curves X{T),
and Fricke subgroups Yq{N) 669
3.7 Modular functions and simple (finite) sporadic groups. The
Monstrous Moonshine. Borcherds theorem 672
4 Dedekind Ca' function for number field K and Seiberg ( function 679
4.1 Algebraic curves (Riemann surfaces) over Q 682
4.2 Algebraic curves X(F) over Q 683
4.3 Eichler Shimura theory 686
4.4 Wiles proof of Last Fermat Theorem 688
4.5 C functions of elliptic Operators on compact Riemann mani
folds. The Seiberg C function 689
4.6 Determinant line bündle associated with family of Dirac op
erators and its Quillen metric 691
4.7 Seiberg C function and trace formula. The length spectrum . 693
Concluding Remarks 697
Suggestions for Further Reading 699
Index 703 |
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| spelling | Maurin, Krzysztof Verfasser aut The Riemann legacy Riemannian ideas in mathematics and physics by Krzysztof Maurin Dordrecht [u.a.] Kluwer Acad. Publ. 1997 XXII, 717 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 417 Aus dem Poln. übers. Riemann, Bernhard <1826-1866> Riemann, Bernhard 1826-1866 (DE-588)118600869 gnd rswk-swf Mathematische Physik Geometry, Riemannian Mathematical physics Riemann, Bernhard 1826-1866 (DE-588)118600869 p DE-604 Mathematics and its applications 417 (DE-604)BV008163334 417 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007954950&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maurin, Krzysztof The Riemann legacy Riemannian ideas in mathematics and physics Mathematics and its applications Riemann, Bernhard <1826-1866> Riemann, Bernhard 1826-1866 (DE-588)118600869 gnd Mathematische Physik Geometry, Riemannian Mathematical physics |
| subject_GND | (DE-588)118600869 |
| title | The Riemann legacy Riemannian ideas in mathematics and physics |
| title_auth | The Riemann legacy Riemannian ideas in mathematics and physics |
| title_exact_search | The Riemann legacy Riemannian ideas in mathematics and physics |
| title_full | The Riemann legacy Riemannian ideas in mathematics and physics by Krzysztof Maurin |
| title_fullStr | The Riemann legacy Riemannian ideas in mathematics and physics by Krzysztof Maurin |
| title_full_unstemmed | The Riemann legacy Riemannian ideas in mathematics and physics by Krzysztof Maurin |
| title_short | The Riemann legacy |
| title_sort | the riemann legacy riemannian ideas in mathematics and physics |
| title_sub | Riemannian ideas in mathematics and physics |
| topic | Riemann, Bernhard <1826-1866> Riemann, Bernhard 1826-1866 (DE-588)118600869 gnd Mathematische Physik Geometry, Riemannian Mathematical physics |
| topic_facet | Riemann, Bernhard <1826-1866> Riemann, Bernhard 1826-1866 Mathematische Physik Geometry, Riemannian Mathematical physics |
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