The Riemann Legacy Riemannian Ideas in Mathematics and Physics by Krzysztof Maurin Division of Mathematical Methods in Physics, University of Warsaw, Warsaw, Poland KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON /LONDON
Contents Foreword: Riemann's Geometric 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 Geodesies in Riemann space (manifold) (M,g) as lines of extremal length. Euler-Lagrange equation..... x. 20 1.6 Jacobi fields (curvature and geodesies) 22 2 Sectional Curvature. Spaces of Constant Curvature. Weyl Hypothesis. x. - 24 2.1 Ovals 28 2.2 Riemannian manifolds as metric spaces (Hopf-Rinow). Geodesic completeness 31 2.3 Symmetric 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 Homplogy. Cohomology. De Rham cohomology... 49 3.2 Hodge theory of harmonic forms 53 v
VI 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 Girding. 64 3.8 Weitzenbrock formulas 67 3.9 Euler form. Hopf theorem on index of vector field 69 3.10 Poincare duality. Kiinneth 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 Chern-Gauss-Bonnet theorem 84 4.1 Allendorfer-Weil formula 87 Curvature and Topology or Characteristic Forms of Chern, Pontriagin, and Euler 90 5.1 Chern forms 90 5.2 Pontriagin forms. Pfaffian R?. Chern theorem once again.. 97 5.3 Hirzenbruch signature theorem 99 5.4 General index theorem (Atiyah-Singer) 100 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..... Ill 6.5 Symmetric spaces 114 6.6 Spectral geometry. 116 II General Structures of Mathematics 119 1 Differentiate Structures. Tangent Spaces. Vector Fields 121 2 Projective (Inverse) Limits of Topological Spaces 134 3 Inductive Limits. Presheaves. Covering Defined by Presheaf 137
Vll 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 Rich Mathematical Structures of Simple Notions of Physics Based on Example of Analytical Mechanics 237 16 Tangent Bundle 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
Vlll 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 7Tk(X, xo). Hopf Fibering. Serre Theorem on Exact Sequence of Homotopy Groups of a Fibering 292 26 Various Benefits of Characteristic Classes (Orientability, Spin Structures). 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 29.12i?-functions. The great Riemann theorems: 'Abel theorem', 'Jacobi inversion', and '# divisor theorem' 357 IV Riemann and Calculus of Variations 361 1 Introduction 363
IX 1.1 General criteria for existence of minimizers of functionals.. 366 1.2 Convexity and weak lower semi continuity 368 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 Teichmiiller Theory. Riemann Moduli Problem 396 3.1 Teichmiiller metric 398 3.2 The analytic structure of the Teichmiiller space T p 398 3.3 The moduli space 399 Riemannian Approach to Teichmiiller Theory. Harmonic Maps and Teichmiiller Space. 401 4.1 Hermitian hyperbolic geometry of Kobayashi.. 414 4.2 Hyperbolic complex analysis 418 4.3. Hyperbolicity of the Teichmiiller space.. :... 418 4.4 Kobayashi pseudodistance. Kobayashi hyperbolic spaces... 419 4.5 Invariant metrics of Teichmiiller 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 Teichmiiller space (general remarks) 432 4.10 The period map. Royden theorems 434 4.11 The period map and Torelli theorems....436 Teichmiiller theory and Plateau-Douglas problem 438 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 problem 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 Hyperkahler quotients 511 9 Direct Methods in Calculus of Variations for Manifolds with Isometries. Equivariant Sobolev Theorems. Yamabe Problem and its Relation to General Relativity 513
XI 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 Hormander 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 regular 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 Flat 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 Ai H ~ E {E) 635 VI Riemann and Number Theory 649 1 Introduction 651 1.1 Introduction 651
Xll 1.2 Automorphic forms, modular functions 653 The Riemann C function 655 2.1 L functions of cusp forms 657 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 theorem... 667 3.5 Hecke theory for congruence subgroups. 668 3.6 Congruence subgroups F C F(l), their modular curves X(F), and Fricke subgroups Fo(iV") 669 3.7 Modular functions and simple (finite) sporadic groups. The Monstrous Moonshine. Borcherds theorem 672 Dedekind K function for number field K and Selberg function 679 4.1 Algebraic curves (Riemann surfaces) over Q... 682 4.2 Algebraic curves X(T) 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 manifolds. The Selberg function 689 4.6 Determinant line bundle associated with family of Dirac operators and its Quillen metric 691 4.7 Selberg function and trace formula. The length spectrum. 693 Concluding Remarks 697 Suggestions for Further Reading 699 Index 703