Riemannian connections: What they are and why we care

This lecture introduces Riemannian connections on manifolds, focusing on their properties and importance. The instructor explains the concept of connections, their symmetry requirements, and compatibility with the metric. The lecture concludes with the fundamental theorem of Riemannian geometry, stating the existence of a unique connection that is both symmetric and compatible with the metric, known as the Riemannian connection or Levi-Civita connection.