Riemannian Geometry rough lecture notes Fall Term 2014 / 15

After recalling some background, we define Riemannian metrics and Riemannian manifolds. We analyze the basic tensorial operations that become available in the presence of a Riemannian metric. Then we construct the Levi-Civita connection, which is the basic " new " differential operator coming from such a metric. Background The purpose of this section is two–fold. On the one hand, we want to relate the general concept of a Riemannian manifold to the geometry of hypersurfaces as known from introductory courses. On the other hand, we recall some facts about tensor fields and introduce abstract index notation.