The Riemann Legacy. Riemannian Ideas in Mathematics and Physics KLUWER ACADEMIC PUBLISHERS. Krzysztof Maurin

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1 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

2 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 : Parallel transport and linear (affine) connection Vector bundles and operations on them Riemann surfaces Riemannian connection. Levi-Civita connection Geodesies in Riemann space (manifold) (M,g) as lines of extremal length. Euler-Lagrange equation..... x Jacobi fields (curvature and geodesies) 22 2 Sectional Curvature. Spaces of Constant Curvature. Weyl Hypothesis. x Ovals Riemannian manifolds as metric spaces (Hopf-Rinow). Geodesic completeness Symmetric spaces Bounded regions in complex plane. Bergman metric (for the first time) 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., ;, ( Homplogy. Cohomology. De Rham cohomology Hodge theory of harmonic forms 53 v

3 VI 3.3 Hodge decomposition The method of heat transport (diffusion equation) The Euler-Poincare characteristic (Euler number) Index theorem (for the first time) Sobolev spaces. Theorems of Rellich, Sobolev, and Girding Weitzenbrock formulas Euler form. Hopf theorem on index of vector field Poincare duality. Kiinneth theorem Intersection number (Kronecker index) of two cycles Index of vector a field and degree of mapping. Kronecker integral Relation between Morse index and index of a vector field.. 82 Chern-Gauss-Bonnet theorem Allendorfer-Weil formula 87 Curvature and Topology or Characteristic Forms of Chern, Pontriagin, and Euler Chern forms Pontriagin forms. Pfaffian R?. Chern theorem once again Hirzenbruch signature theorem General index theorem (Atiyah-Singer) 100 Kahler Spaces. Bergman Metrics. Harish-Chandra-Cartan Theorem. Siegel Space (once again!) Calabi hypothesis and Calabi-Yau spaces Bergman metrics on bounded domains Imbedding in projective spaces. Kodaira theorem Homogeneous complex spaces and bounded domains..... Ill 6.5 Symmetric spaces Spectral geometry. 116 II General Structures of Mathematics Differentiate Structures. Tangent Spaces. Vector Fields Projective (Inverse) Limits of Topological Spaces Inductive Limits. Presheaves. Covering Defined by Presheaf 137

4 Vll 4 Algebras. Groups, Tensors, Clifford, Grassmann, and Lie Algebras. Theorems of Bott-Milnor, Wedderburn, and Hurwitz Fields and their Extensions Galois Theory. Solvable Groups Ruler and Compass Constructions. Cyclotomic Fields. Kronecker- Weber Theorem Algebraic and Transcendental Elements Weyl principle Topology of Compact Lie Groups Representations of Compact Lie Groups Nilpotent, Semimple, and Solvable Lie Algebras Reflections, Roots, and Weights. Coxeter and Weyl groups Weights of representations of Lie algebra Classification of root systems. Coxeter diagrams Relation with semisimple complex Lie algebras Covariant Differentiation. Parallel Transport. Connections Remarks.on Rich Mathematical Structures of Simple Notions of Physics Based on Example of Analytical Mechanics Tangent Bundle TM. Vector, Fiber, Tensor and Tensor Densities, and Associate Bundles G-spaces. Group Representations Principal and Associated Bundles Induced Representations and Associated Bundles Vector Bundles and Locally Free Sheaves Axiom of Covering Homotopy 271

5 Vlll 22 Serre Fibering. General Theory of Connection. Corollaries Homology. Cohomology. de Rham Cohomology Cohomology of Sheaves. Abstract de Rham Theorem Homotopy Group 7Tk(X, xo). Hopf Fibering. Serre Theorem on Exact Sequence of Homotopy Groups of a Fibering Various Benefits of Characteristic Classes (Orientability, Spin Structures). Clifford Groups, Spin Group Divisors and Line Bundles. Algebraic and Abelian Varieties General Abelian Varieties and Theta Function Theta functions, Strictly transcendental extensions. Transcendental degree Theorems on Algebraic Dependence 318 III The Idea of the Riemann Surface Introduction Fredholm-Noether operators. Parametrices Proof of Riemann-Roch theorem The fundamental theorem for compact surfaces Embedding of Riemann surfaces Hyperelliptic surfaces. Hyperelliptic involutions Weierstrass points. Wronskian Hyperelliptic involution : Clifford theorem ' Riemann bilinear relations. Abel-Jacobi map Linear bundles on complex tori: Appel-Humbert theorem i?-functions. The great Riemann theorems: 'Abel theorem', 'Jacobi inversion', and '# divisor theorem' 357 IV Riemann and Calculus of Variations Introduction 363

6 IX 1.1 General criteria for existence of minimizers of functionals Convexity and weak lower semi continuity 368 The Plateau Problem Coercity of Dirichlet integral The Rado-Douglas solution of Plateau problem Riemann mapping theorem and Plateau problem Representation formulas for minimal surfaces. Enneper- Weierstrass theorem. Scherk surface Minimal surfaces and value distribution theory Some properties of harmonic maps. Theorems of Eells- Sampson, Hartman, and corollaries 388 Teichmiiller Theory. Riemann Moduli Problem Teichmiiller metric The analytic structure of the Teichmiiller space T p The moduli space 399 Riemannian Approach to Teichmiiller Theory. Harmonic Maps and Teichmiiller Space Hermitian hyperbolic geometry of Kobayashi Hyperbolic complex analysis Hyperbolicity of the Teichmiiller space.. : Kobayashi pseudodistance. Kobayashi hyperbolic spaces Invariant metrics of Teichmiiller space Harmonic Beltrami differentials on (M, g) Wolpert formulas for Petersson-Weil form Generalization to higher dimensions Metrics on Teichmiiller space (general remarks) The period map. Royden theorems The period map and Torelli theorems Teichmiiller theory and Plateau-Douglas problem 438 Rescuing Riemann's Dirichlet Principle. Potential Theory Subharmonic functions." Riesz decomposition Poisson integral and Harnack theorems History of the potential theory ' Perron method

7 6.5 Rado theorem. Theorem of Poincare-Volterra The Royal Road to Calculus of Variations (Constantin Caratheodory) Introduction Fields An equivalent problem Integrability conditions. Geodesic fields. (Independent) Hilbert integral Weierstrass excess function and condition for strong minimum Legendre condition for weak minimum Complete figure of variational problem Problems with free endpoints. Broken extremals Legendre transformation. Canonical equations of Hamilton. Hilbert integral in canonical coordinates. Hamilton Jacobi theory Physical meaning of functions H, S, and L Lagrange bracket and geodesic fields Canonical transformations Caustics. 'Enveloppensatz' of Caratheodory. Singularities Finsler geometry and geometric optics General Huygens principle and Finsler geometry 479 7:16 Field theories for calculus of variation for multiple integrals Lepage theory of geodesic fields Caratheodory and thermodynamics (second law). Pfaff problem and Frobenius theorem Caratheodory and the beginning of calculus of variations' Symplectic and Contact Geometries. Conservation Laws Introduction Lie approach to hamiltonian mechanics Conservation laws and 'Postulates of impotence' Momentum map and symplectic reduction. (Reduction of phase space for systems with symmetries) Hyperkahler quotients Direct Methods in Calculus of Variations for Manifolds with Isometries. Equivariant Sobolev Theorems. Yamabe Problem and its Relation to General Relativity 513

8 XI V Riemann and Complex Geometry Introduction On Complex Analysis in Several Variables Ellipticity, Runge Property, and Runge Type Theorems Hormander Method in Complex Analysis Wirtinger Theorems. Metric Theory of Analytic Sets The Problem of Poincare and the Cousin Problems Ringed Spaces and General Complex Spaces % Construction of Complex Spaces by Gluing and by Taking Quotient Construction of complex spaces by gluing '.'... : On deformations of regular families of complex structures (Grauert theory) Grauert solution of main problems of deformation theory of complex structures On differential calculus on complex spaces From Riemann period relations to theorems of Kodaira and Grauert Concluding remarks Differential Geometry of Holomorphic Vector Bundles over Compact Riemann Surfaces and Kahler manifolds. Stable Vector Bundles, ' Hermite-Einstein Connections, and their Moduli Spaces Flat bundles and flat connections Moduli spaces of H-E structures Hermite-Einstein metrics (structures) as critical points of Donaldson functional (variational theory of H-E connections) Kahler structures on moduli space Ai H ~ E {E) 635 VI Riemann and Number Theory Introduction Introduction 651

9 Xll 1.2 Automorphic forms, modular functions 653 The Riemann C function L functions of cusp forms 657 Hecke Theory Petersson Scalar Product '.'.." Hecke operators Hecke L series Ramanujan-Petersson conjecture and Deligne theorem Hecke theory for congruence subgroups Congruence subgroups F C F(l), their modular curves X(F), and Fricke subgroups Fo(iV") Modular functions and simple (finite) sporadic groups. The Monstrous Moonshine. Borcherds theorem 672 Dedekind K function for number field K and Selberg function Algebraic curves (Riemann surfaces) over Q Algebraic curves X(T) over Q Eichler-Shimura theory Wiles proof of Last Fermat Theorem C functions of elliptic operators on compact Riemann manifolds. The Selberg function Determinant line bundle associated with family of Dirac operators and its Quillen metric Selberg function and trace formula. The length spectrum. 693 Concluding Remarks 697 Suggestions for Further Reading 699 Index 703

Modern Geometric Structures and Fields

Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface