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 de Rham cohomology, integration and Stokes theorem.  

Aim of the course

Provide an introduction to vector bundles, connections, geodesics, and the first steps on characteristics classes.

Concepts covered in the course will include:

1. Vector bundles and connections,

2. Short introduction to Cech cohomology,

3. Curvature, Bianchi identity, first Chern class, Euler class and Thom class,

4. Parallel transport,

5. Connections on the tangent bundle and geodesics,

6. The exponential map.

Some of the results we will cover include:

1. Classification of line bundles,

2. Tubular neighbourhood theorem,

3. A weak version of the Jordan closed curve theorem,

3. Geodesics as length minimizing curves,

4. Gauss--Bonnet theorem...