Differential manifolds are geometric objects locally described by coordinate systems (real or complex), but possessing a global structure that can be particularly intricate. These objects arise naturally in mathematics and also serve as the framework upon which many Physics theories are built (general relativity, gauge theories, etc.).
This course is divided into two parts.
The first part is an introduction to differential manifolds and some of the key concepts associated with them (charts, atlases). We will start with the definition of abstract manifolds possessing a differential structure and provide examples of such manifolds, along with some recipes for constructing differential manifolds (quotient manifolds, connected sums, submanifolds, etc.). We will particularly focus on constructing 2-dimensional manifolds, which will give us a first glimpse of the complexity of manifolds. Once this general framework is established, we will introduce the notion of smooth maps between manifolds, as well as the concepts of embedding and submersion.
We will study the consequences of Sard's Theorem, which serves as the starting point for several developments, including Whitney's embedding theorems, which implies that (abstract) manifolds are not fundamentally more general than the simpler notions of submanifolds of Euclidean space. Sard's Theorem also forms the basis of the theory of transversality and intersection theory, which will lead us to the notion of the degree of a map defined between manifolds and the definition of the Euler characteristic of a manifold.
All these tools will give us the opportunity to approach Morse Theory, which allows for a better understanding of the topological complexity of a compact manifold by studying the critical points of functions. From this, we will derive Reeb's Theorem, which characterizes spheres.
Next, we will define and study bundles and sections of bundles, which are ubiquitous in geometry as well as in various physics theories. We will delve deeper into vector bundles, including the tangent bundle, cotangent bundle, and normal bundle. Studying the Hopf fibration (and its generalizations in higher dimensions) will help us better understand the challenges related to the classification of bundles.
Differential manifolds provide the natural framework for a rich integration theory. This theory, which relies on the notions of differential forms and exterior derivative, will provide us with a unified framework for defining the gradient, divergence, and curl operators, as well as the associated integration by parts formulas (Stokes' Theorem). We will also introduce some basics of de Rham cohomology, connecting with the course on algebraic topology.
The second part of the course builds upon the concepts developed in the first part. We will begin by defining the notion of a metric (Riemannian or pseudo-Riemannian) on a differential manifold. We will then introduce the Levi-Civita connection as an operator acting on sections of the tangent bundle, leading us to the notions of parallel transport and curvature (Riemann curvature tensor). We will define and study geodesics on a Riemannian (or pseudo-Riemannian) manifold, which will allow us to define geometric normal coordinates on a manifold.
In the context of Riemannian manifolds, we will define various notions of curvature: Riemann curvature tensor, sectional curvature, Ricci curvature tensor, and scalar curvature. We will particularly focus on the case of surfaces embedded in 3-dimensional Euclidean space, which will give us a better understanding of the notions of connection and curvature. Finally, we will state and prove some results regarding the links between curvature (especially its sign), topology, and geometry of Riemannian manifolds, such as the (Chern-)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
- Teaching coordinator: Pacard Frank