Location and time:

Aim:
The aim of this course is to give a thorough introduction to Riemannian geometry and the concept of holonomy, motivated by application to Calabi-Yau manifolds of string theory.

Course description:
The course starts with a discussion of vector bundles on manifolds, the notion of connection on a vector bundle and the associated notions of covariant differentiation and of paralleltransport along a curve. Paralleltransport along a closed curve need not be the identity; this gives rise to the notion of holonomy. Infinitesimally, holonomy is measured by the so-called curvature tensor.
Parallel to this discussion, 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.

A Riemannian metric gives rise to a uniquely defined connection on the tangent bundle, the so-called Levi-Civita connection. The associated notions of Riemannian curvature, sectional curvature, Ricci curvature and scalar curvature will be discussed.
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.
In the second half of the course we introduce the notion of almost complex structures and complex structures on a Riemannian manifold. This leads to the definitions of Hermitean and complex manifolds, and later to Kahler manifolds which will be discussed in more detail. Finally, we discuss the holonomy groups and curvature properties of Calabi-Yau spaces. We focus mostly on Calabi-Yau threefolds, since they play an important role in string theory compactifications.

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:

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. Basic knowledge of differentiable manifolds, submersions, immersions, vector fields, as described in Prerequisites from differential geometry by E.P. van den Ban. In case of insufficient background knowledge, the teachers will suggest reading material to make up for it.