Distance, geodesics and complete manifolds: Complete manifolds

This lecture covers the concept of distance, geodesics, and complete manifolds. It introduces the Riemannian distance on a manifold, defining it as a metric space. The lecture explains the relationship between metric topology and manifold topology, emphasizing the existence of minimizing geodesics. It discusses Cauchy sequences, metric completeness, and the Hopf-Rinow theorem, highlighting the equivalence between geodesic completeness and metric completeness on a connected manifold. The lecture concludes with an example illustrating the completeness of a manifold embedded in Euclidean space.