Les variétés différentielles sont des objets géométriques localement décrits par des systèmes de coordonnées (réelles ou complexes), mais dotés de structures globales pouvant être extrêmement riches, voire complexes. Ces objets apparaissent naturellement en mathématiques et constituent également le cadre de nombreuses théories physiques comme la relativité générale ou les théories de jauge.
Le cours débute par une introduction aux variétés différentielles et aux concepts fondamentaux qui leur sont associés : cartes, atlas, structures différentielles. Nous donnerons la définition rigoureuse d’une variété (abstraite), illustrée par de nombreux exemples, et nous présenterons quelques méthodes de construction de variétés : variétés quotient, sous-variétés, sommes connexes, etc. L'accent sera mis sur les surfaces (i.e. les variétés de dimension 2), qui offrent un premier aperçu de la grande diversité des phénomènes topologiques et géométriques.
Nous introduirons ensuite les notions d’applications lisses entre variétés, de plongement et de submersion. Cela nous conduira naturellement au théorème de Sard (i.e., 2-dimensional manifolds), which offer an accessible entry point into the diversity of topological and geometric phenomena.
We will then introduce the notions of smooth maps between manifolds, immersions, and submersions. This will naturally lead us to Sard’s Theorem, a cornerstone result with far-reaching consequences in mathematics. Sard’s Theorem plays a central role in transversality theory and intersection theory, and it allows us to define key topological invariants such as the degree of a map and the Euler characteristic of a manifold.
We will then turn to the notion of fiber bundles, which are ubiquitous in both geometry and theoretical physics. A substantial part of the course will be devoted to the study of vector bundles, such as the tangent, cotangent, and normal bundles. The classical example of the Hopf fibration will serve as an opportunity to explore the subtleties and challenges related to the classification of bundles.
Differential manifolds also provide the natural setting for a unified theory of integration, based on differential forms, the exterior derivative, and the wedge product. This formalism recasts classical vector calculus operators — gradient, divergence, and curl — in geometric terms, and leads to integral formulas that culminate in the generalized Stokes theorem, which unifies several fundamental results.
With this foundation in place, we will introduce the notion of a metric (Riemannian or Lorentzian) and that of a connection on a vector bundle. In particular, we will study the Levi-Civita connection, which allows one to differentiate sections of the tangent bundle. This leads to the study of parallel transport, followed by geodesics, which in turn provide a geometric construction of normal coordinates around a point.
The Levi-Civita connection will then allow us to define and explore various notions of curvature on a manifold: the Riemann curvature tensor, sectional curvature, Ricci curvature, and scalar curvature. To gain a more concrete understanding of these objects, we will study in detail the case of surfaces embedded in three-dimensional Euclidean space.
Finally, we will establish several key results linking curvature, topology, and geometry, such as the Gauss–Bonnet Theorem and Myers’ Theorem.
Bibliography :
- Manfredo P. do Carmo : Riemannian Geometry, Birkhäuser.
- Victor Guillemin and Alan Pollack : Differential Topology. Prentice-Hall.
- Morris W. Hirsh : Differential Topology, Springer.
- Peter Petersen : Riemannian Geometry, Springer.
Course language: Textbook in english, Lectures in French or English
ECTS credits: 5
