Riemannian connections

This lecture covers the concept of Riemannian connections on manifolds, defining connections as maps that relate tangent vectors on a manifold. It explores the properties and operations of these connections, emphasizing their smoothness and compatibility with the metric. The lecture also delves into the Leibniz rule and the symmetric nature of connections. Additionally, it discusses the unique Riemannian connection, known as the Levi-Civita connection, which is both symmetric and metric-compatible. The presentation concludes by highlighting the existence of infinitely many connections on a manifold and the significance of these connections in Riemannian geometry.