Differentiable manifolds are geometric objects, locally parametrized by coordinate systems, but with a global topology that can be nontrivial. They are therefore the natural language of differential geometry (Riemannian, symplectic, complex, etc...), and also of many physical theories (general relativity, gauge theory, etc...).
The goal of this course is to provide an introduction to manifolds, and to a number of related key concepts: vector bundles, differential forms, de Rham cohomology.
Being by definition locally modeled on vector spaces, manifolds give rise to an elegant formalism for differential calculus, allowing to study maps between manifolds by linearization. This will lead us to the famous Whitney embedding theorem, showing that "abstract" manifolds are no more general than the more familiar submanifolds of Euclidian space.
Perhaps more surprinsingly, manifolds are also the natural setting for an integration theory of infinitesimal elements of length, area, volume, etc.., embodied in the notion of differential form. The corresponding formalism is here again beautifully concise, summing up in one stroke the classical gradient, divergence and curl operators from fluid mechanics and electromagnetism, and the integration-by-parts formula they satisfy.
The construction of differential forms relies on the similarly fruitful concept of vector bundles, which is for instance at the heart of gauge theory. We will introduce the basics of this theory, which consists in doing (multi)linear alebra on a smooth family of vector spaces.
Finally, we will take a few steps into de Rham cohomology, a device which encodes through (linear) homological algebra the idea that "holes" in a space can be detected by integrating around them, offering a first encounter with algebraic topology.
Required level: A familiarity with the first part of the MAT431 course (of an euclidian space), it is not mandatory but keenly suggested.
Bibliography :
- Lee : Introduction to smooth manifolds
- Milnor : Topology from the differentiable viewpoint.
- Bott et Tu : Differential forms in algebraic topology
Course language: French or English
ECTS credits: 5
- Teaching coordinator: Pacard Frank