This is an introduction to Riemannian manifolds. We'll cover concepts such as distance, volume, covariant derivatives, the Levi-Civita connection, geodesics, different notions of curvature: sectional, Ricci, scalar; Jacobi fields, conjugate points and will build towards proving the Rauch comparison, Hopf-Rinow, and volume comparison theorems.
We'll use several sources including: Riemannian Geometry by Gallot, Hulin, and LaFontaine, Riemannian Geometry by Sakai, and Riemannian Geometry by Peter Petersen.
Advanced Calculus. Some elementary set theory and metric topology.