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
The aim of this course is to provide an introduction to bundles,
connections and curvature. These are important notions in contemporary
differential geometry, and their theory provides some fundamental
techniques that are widely used in the field.
More specifically, the course will cover the following topics:
- fiber bundles, principal bundles and vector bundles
- connections, curvature
- Characteristic classes
- Riemannian geometry, connections and geodesics.
For the part on principal bundles, we need some of the theory of Lie groups. We will cover the necessary parts (without proofs) in this lecture. Following the course on Lie groups in parallel to this course will provide more background.
During the course we will touch upon applications in and connections to other field of mathematics (e.g. topology) and physics. As a final result we will give a proof the Chern-Gauss-Bonnet theorem, which is a higher dimensional generalization of the Gauss-Bonnet theorem linking curvature to topology.
Hessel Posthuma (UvA)