Géométrie différentielle

Contents

Géométrie différentielle is the mathematical study of geometric structures on smooth manifolds, combining techniques from Carl Friedrich Gauss, Bernhard Riemann, Élie Cartan, Henri Poincaré and later contributors such as Shiing-Shen Chern, Marston Morse, Eugenio Beltrami and S. S. Chern to analyze curvature, connections and global invariants. It arose from problems in classical Differential calculus and Integral calculus and was shaped by developments in General relativity, Gauge theory, Global analysis and the classification work of Andrey Kolmogorov and John Milnor. Modern treatments interrelate with research programs at institutions like Institut des Hautes Études Scientifiques, University of Göttingen, Princeton University and École Normale Supérieure.

Introduction

Géométrie différentielle studies smooth structures on manifolds introduced by Bernhard Riemann and formalized by Hermann Weyl, emphasizing curvature via the work of Carl Friedrich Gauss and Élie Cartan. It connects to classical problems addressed in treatises by Leonhard Euler and later modernized through lectures by Henri Poincaré and texts from William Thurston. The field underpins physical theories developed by Albert Einstein and mathematical frameworks advanced by Michael Atiyah, Isadore Singer and Simon Donaldson.

Foundational notions include smoothness drawn from Augustin-Louis Cauchy and Karl Weierstrass-influenced analysis, coordinate charts inspired by Riemann's habilitation, and topological constraints studied by Poincaré. Key concepts are differentiable manifolds related to classification results of John Milnor and William Thurston, tangent spaces connected to linear algebra from Arthur Cayley and Hermann Grassmann, and curvature tensors developed by Riemann and expanded by Elie Cartan and Weyl. Global invariants such as characteristic classes owe to work by Shiing-Shen Chern, Raoul Bott and Atiyah–Singer enthusiasts like Michael Atiyah and Isadore Singer.

A differentiable manifold is defined via atlases of charts influenced by Bernhard Riemann and formalized in modern topology by Poincaré and Seifert. Transition maps reflect analytic rigor pioneered by Karl Weierstrass and Émile Picard, while existence theorems for structures draw on results by John Nash and embedding theorems connected to Stephen Smale and Mikhail Gromov. Constructions of exotic manifolds reference discoveries of Milnor and subsequent work at Institute for Advanced Study and Université Paris-Sud, with classification problems informed by contributions from Freedman and Donaldson.

Vector fields and differential forms follow differential calculus methods from Newton and are formalized using exterior algebra developed by Hermann Grassmann and Élie Cartan. The Lie derivative and flows relate to dynamical viewpoints in Henri Poincaré's qualitative theory, while exterior differentiation and de Rham cohomology trace to Georges de Rham and link to index theorems by Atiyah and Singer. Connections and covariant derivatives were systematized by Élie Cartan and later used by Yang–Mills theorists such as Chen Ning Yang and Robert Mills; curvature forms appear in treatments by Shiing-Shen Chern and in characteristic classes by Raoul Bott and J. H. C. Whitehead.

Riemannian geometry studies metrics introduced by Bernhard Riemann and developed through curvature concepts by Carl Friedrich Gauss and global comparison techniques from Aleksandr Aleksandrov and Marston Morse. The study of geodesics is central to work by Levi-Civita and applications in relativity by Albert Einstein; Ricci flow methods were pioneered by Richard Hamilton and used by Grigori Perelman in geometrization problems related to William Thurston. Spectral geometry connects to conjectures and theorems studied by Mark Kac, Atiyah, and Patodi, while Sobolev and elliptic estimates trace analytic techniques to Sergei Sobolev and Leray.

Applications span theoretical physics—General relativity, Yang–Mills theory, and String theory—and pure mathematics including Topology, Complex geometry influenced by Henri Poincaré and André Weil, and algebraic techniques developed by Alexander Grothendieck and Jean-Pierre Serre. Contemporary interactions include gauge-theoretic invariants from Simon Donaldson and Edward Witten, index theory of Atiyah–Singer with consequences in K-theory explored by Max Karoubi, and geometric analysis informing problems addressed at Courant Institute and Massachusetts Institute of Technology. Computational and applied strands connect to numerical relativity teams at CERN and modeling initiatives at NASA and industrial collaborations with research groups at CNRS and INRIA.