Location and time:

Aim:
The aim of this course is to give a thorough introduction to differential geometry, starting with the basics, and ending with the Gauss Bonnet theorem for surfaces.

Course description:
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 described 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 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. Description: 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.

Organization:
Each meeting will consist of 2 lectures of 45 minutes followed by an assisted exercise session. Each student is allowed to hand in one of the weekly exercises, marked by a star, for correction. This should be done on the date under which the exercise is listed in Exercises.

Examination:
The course will be concluded by

Literature: We will use lecture notes of E. Looijenga. At this moment these notes are available in a Dutch version. An English version is in progress.

Recommended side reading:

Prerequisites:
A good knowledge of multi-variable calculus: chain rule, inverse and implicit function theorem; substitution rule for integration, classical theorems of Gauss and Stokes. Basic knowledge of tensor products of linear spaces and the associated multilinear algebra, such as Notes on tensors, Sect. 1-4. In case of insufficient background knowledge, the teachers will suggest reading material to make up for it.