Differential Geometry (GQT)

 After discussing the basic definitions of manifold theory,
vector bundles and differential forms, the notion of a Riemannian manifold will be introduced. This is a manifold equipped with a Riemannian metric, i.e., a positive definite inner product on the tangent space at each point, depending smoothly  on that point.
The Riemannian metric allows the definition of length of a smooth curve. A curve that is locally of shortest length, is called a geodesic. In local coordinates, a  geodesic may be descibed in terms of a second order ordinary differential equation. The geodesic equation is best described in terms of covariant differentiation, i.e., differentiation with respect to a so-called connection on the bundle of tangent spaces.
More generally we will discuss the notion of connection on a vector bundle, the associated parallel transport, and the curvature tensor.
A Riemannian metric gives rise to a uniquely defined
connection, the so-called Levi-Civita connection. The associated notions of sectional curvature and scalar curvature will be discussed. The primary goal is to give a proof of the Gauss-Bonnet theorem for a compact Riemannian surface, which links the scalar curvature to the topology of the manifold. 
If time permits, another application of the theory will be given.

The course will be concluded by a written exam. A student has two options: 
1) to have his final grade based on both written exam (50  %) and starred exercises (50 %).
 2) to have his final grade entirely based on the written exam.