x Contents 3 Some Remarks on A Contribution to Electrodynamics by Bernhard Riemann Hubert Goenner 1 Introduction Riemann s New Result

Transcription

1 Contents Preamble... 1 Looking Backward: From Euler to Riemann Introduction Functions Elliptic Integrals Abelian Functions Hypergeometric Series The Zeta Function On Space Topology Differential Geometry Trigonometric Series Integration Conclusion References Part I Mathematics and Physics 2 Riemann on Geometry, Physics, and Philosophy Some Remarks Jeremy Gray 1 Introduction The Hypotheses Influences Heat Diffusion and the Commentatio References ix

2 x Contents 3 Some Remarks on A Contribution to Electrodynamics by Bernhard Riemann Hubert Goenner 1 Introduction Riemann s New Result of 1858: The Retarded Potential Gauss, Weber, and Riemann on Electrodynamic Interaction Riemann s Paper Concluding Remarks References Riemann s Memoir Über das Verschwinden der #-Functionen Christian Houzel 1 Jacobi s Inversion Problem A Crucial Observation on Theta Functions The First Step of Riemann s Proof The Second Step of Riemann s Proof The Conclusion of the Proof Later Developments References Riemann s Work on Minimal Surfaces Sumio Yamada 1 Introduction On the Surface of Least Area with a Given Boundary Representation Formulas by Riemann and Weierstrass-Enneper Closing Remarks References Physics in Riemann s Mathematical Papers Introduction Function Theory and Riemann Surfaces Riemann s Memoir on Trigonometric Series Riemann s Habilitationsvortrag 1854 Space and Matter The Commentatio and the Gleichgewicht der Electricität Riemann s Other Papers Conclusion References Cauchy and Puiseux: Two Precursors of Riemann Introduction Algebraic Functions and Uniformization Puiseux and Uniformization

3 Contents xi 4 Cauchy and His Work on Functions of a Complex Variable Uniformization Again References Riemann Surfaces: Reception by the French School Introduction Riemann Surfaces The Nineteenth-Century French Treatises on Analysis Simart s Dissertation Other French Dissertations and Other Works of Riemann On the Relations Between the French and German Mathematicians In a Way of Conclusion References Part II Philosophy 9 The Origin of the Notion of Manifold: From Riemann s Habilitationsvortrag Onward Ken ichi Ohshika 1 Introduction Kantian Worldview Riemann s Habilitationsvortrag Poincaré s Analysis Situs Definitions Using Local Charts According to Hilbert, Weyl, Kneser and Veblen-Whitehead Conclusion: Philosophical Significance References Deleuze et la Géométrie Riemannienne: Une Topologie des Multiplicités Franck Jedrzejewski 1 Introduction Variété et multiplicité Espaces, mesures et multiplicités Typologies des multiplicités Conclusion Extended English Abstract References

4 xii Contents 11 Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts Arkady Plotnitsky 1 Introduction Philosophy: Planes of Thought and the Architecture of Concepts Mathematics: Space, Geometry, and the Concept of Manifold Physics: The Reality Underlying Space Conclusion References Part III Some Recent Developments 12 The Riemann Mapping Theorem and Its Discrete Counterparts Feng Luo 1 Introduction Koebe Andreev Thurston s Circle Packing Theorem A Discrete Uniformization Theorem References The Riemann Roch Theorem Norbert A Campo, Vincent Alberge and Elena Frenkel 1 Introduction Line Bundles Sheaf Cohomology Further Preparations The Riemann Roch Theorem Divisors and the Riemann Roch Theorem The Use of the Riemann Roch Theorem in Teichmüller s Work References Metric Geometries in an Axiomatic Perspective Victor Pambuccian, Horst Struve and Rolf Struve 1 Introduction Metric Planes Higher-Dimensional Metric Spaces The Dimension-Free Case Projective-Metric Geometry Cayley-Klein Geometries References

5 Contents xiii 15 Generalized Riemann Sums Toshikazu Sunada 1 Introduction Generalized Riemann Sums Classical Example Classical Example The Inclusion-Exclusion Principle Generalized Poisson Summation Formulas Is Z d prim a Quasicrystal? References From Riemannian to Relativistic Diffusions Jacques Franchi 1 Introduction Euclidean Brownian Motion Riemannian Brownian Motion The Relativistic Dudley Diffusion in Minkowski Space The Lorentzian Frame Bundle GðMÞ over ðm; gþ The Basic Relativistic Diffusion Covariant N-relativistic Diffusions Example of Robertson-Walker (R-W) Manifolds Sectional Relativistic Diffusion References Part IV Relativity 17 On the Positive Mass Theorem for Closed Riemannian Manifolds Andreas Hermann and Emmanuel Humbert 1 Introduction ADM Mass in General Relativity The Mass of a Closed Manifold Equivalence of the Two Positive Mass Conjectures Some Recent Results on the Positive Mass Conjecture An Idea of the Proof of Theorem Preservation of Mass by Surgery References On Local Characterization Results in Geometry and Gravitation Marc Mars 1 Introduction Classical Characterizations

6 xiv Contents 3 Local Characterizations of the Schwarzschild and Kruskal Spacetimes Local Characterization of pp-waves and Related Spacetimes Local Characterizations of the Kerr, Kerr Newman and Kerr De Sitter Metrics References The Conformal Approach to Asymptotic Analysis Jean-Philippe Nicolas 1 Introduction Conformal Compactification Peeling Conformal Scattering Concluding Remarks References Part V Concluding Chapter 20 Bernhard Riemann and His Work Lizhen Ji 1 Introduction Riemann s Work I: His Best Known Works Riemann s Work II: Some Little Known or Even Unknown Works Riemann s Publications and his Impact How Riemann Developed People Who Influenced Riemann References Index

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