Tangent Bundles and Vector Fields

This lecture introduces the concept of smooth maps between manifolds and defines the differential of a smooth map. It explores the notion of vector fields on manifolds, relating them to gradients in Euclidean space. The lecture also covers retractions, which are used to move around manifolds by providing a way to smoothly follow curves. The instructor explains the definition of a retraction as a smooth map from the tangent bundle to the manifold, ensuring that it captures the idea of smoothly varying curves with respect to the manifold. The lecture concludes by discussing the properties of retractions and their role in defining smooth vector fields.