Riemannian Geometry

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.