From Riemann to Differential Geometry and Relativity

1. Looking Backward: From Euler to Riemann
Athanase Papadopoulos

Part I. Mathematics and Physics

2. Riemann on Geometry, Physics, and Philosophy—Some Remarks
Jeremy Gray

3. Some Remarks on “A Contribution to Electrodynamics” by Bernhard Riemann
Hubert Goenner

4. Riemann’s Memoir Über das Verschwinden der



Christian Houzel

5. Riemann’s Work on Minimal Surfaces
Sumio Yamada

6. Physics in Riemann’s Mathematical Papers
Athanase Papadopoulos

7. Cauchy and Puiseux: Two Precursors of Riemann
Athanase Papadopoulos

8. Riemann Surfaces: Reception by the French School
Athanase Papadopoulos

Part II. Philosophy

9. The Origin of the Notion of Manifold: From Riemann’s Habilitationsvortrag Onward
Ken’ichi Ohshika

10. Deleuze et la Géométrie Riemannienne: Une Topologie des Multiplicités
Franck Jedrzejewski

11. Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts
Arkady Plotnitsky

Part III. Some Recent Developments

12. The Riemann Mapping Theorem and Its Discrete Counterparts
Feng Luo

13. The Riemann–Roch Theorem
Norbert A’Campo, Vincent Alberge, Elena Frenkel

14. Metric Geometries in an Axiomatic Perspective
Victor Pambuccian, Horst Struve, Rolf Struve

15. Generalized Riemann Sums
Toshikazu Sunada

16. From Riemannian to Relativistic Diffusions
Jacques Franchi

Part IV. Relativity

17. On the Positive Mass Theorem for Closed Riemannian Manifolds
Andreas Hermann, Emmanuel Humbert

18. On Local Characterization Results in Geometry and Gravitation
Marc Mars

19. The Conformal Approach to Asymptotic Analysis
Jean-Philippe Nicolas

Part V. Concluding Chapter

20. Bernhard Riemann and His Work
Lizhen Ji

Keywords: Mathematics, History of Mathematical Sciences, Differential Geometry