M1: Differential Geometry - 8EC

Aim of the course

Provide an introduction to vector bundles (with application to tubular neighborhoods), principal bundles, connections, the general theory of geometric structures (G-structures) and their integrabiliy, mentioning examples such as Riemannian, complex or symplectic structures.

The course will have several parts:

1. One which concentrates on vector-bundles and connections (including parallel transport, curvature and the construction of the first Chern class). Here we will also discuss the tubular neighborhood theorem.

2. One which concentrates on principal bundles and connections, and where we explain that, for principal $GL_n$-bundles, the resulting theory is equivalent to the one for vector bundles. This part will start with 1-2 lectures about the basic notions/facts from Lie groups that are needed here.

3. While along the way we will mention some examples of geometric structures (such as metrics), in the last parts of the course we will concentrate on a general framework that allows one to treat many geometric structures in an unified way: the framework provided by the theory of $G$-structures. Here we will present the framework and examples such as: Riemannian metrics, distributions, foliations, symplectic structures, almost complex and complex structures.

4. Finally, in the last two lectures, we will concentrate on the integrability of G-structures and the torsion of G-structures as obstruction to integrability. For instance, for symplectic structures: one talks about almost symplectic structures (non-degenerate two forms) and their integrability is about the form being closed (Darboux theorem); for foliations one talks about sub-bundle of tangent bundle, and their integrability is equivalent to the involutivity (Frobenius theorem); etc etc.

Prerequisites

  • a good knowledge of multi-variable calculus
  • some basic knowledge of topology (such as compactness)
  • the standard basic notions that are taught in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a vector field, the tangent space (and bundle), the definition of differential forms, de Rham operator (and hopefully the definition of de Rahm cohomology)
  • some very basic knowledge of Lie theory may be useful, but it is not required.