A Riemannian manifold is a manifold with a smooth choice of a scalar product on its tangent spaces (called a Riemannian metric). In this course, we will study some of the numerous concepts arising from this innocent looking extra piece of structure on a manifold. Here is an incomplete list of topics which will appear in the course (not necessarily in this order):

  • We will define a distance funtion on the manifold using the Riemannian metric which makes the manifold into a metric space. We will discuss geodesics, which are the suitable generalisations of straight lines in the euclidean space, and geodesic completeness. The Hopf-Rinow theorem relating geodesic completeness and metric completeness will be proved.
  • After a brief introduction to connections on vector bundles, we will prove the fundamental Theorem of Riemannian geometry, which states the existence of a connection, the so called Levi-Civita connection, with special properties. The Levi-Civita connection allows us to somewhat canonically identify tangent spaces at different points isometrically (parallel tansport) and, as a result, to differentiate tensor fields on the manifold along vector fields (covariant differentiation).
  • We will use the Levi-Civita connection to define one of the most important invariants of the Riemannian structure: the Riemannian curvature tensor. We will talk about its fundamental properties and the notions of Ricci and scalar curvature which can be extracted from it.
  • We will discuss some of the ways in which the curvature tensor of a Riemannian manifold can be used to obtain information about its topology (e.g. Synge and Myers' Theorem)
  • The Riemannian metric gives rise to a special measure on the manifold and we can thus talk about volumes of (Borel measurable) subsets. We will discuss how and when the Ricci and scalar curvature give us volume estimates. Furthermore, we will talk about a geomtric interpretation of scalar curvature obtained by looking at the volumes of small geodesic balls.
  • Classification resuts for spaces with constant (sectional) curvature

The participants are expected to know the basics of multilinear algebra and analysis on manifolds. A primer containing a brief summary of the necessary background material will be provided in moodle before the lecture starts.