We will give an introduction to Riemannian geometry. We will start with the notion of a manifold, tangent bundles and cotangent bundles. We then introduce Riemannian tensor, Levi-Civita connection, exponential maps, curvature tensor etc.
After establishing the above basic notion and properties, we will introduce Bochner formula and Ricci curvature comparison and applications. We will also introduce the Gromov-Hausdorff topology on the set of isometric classes of all compact metric spaces.