Manifolds and Tangent Space

This lecture covers the concept of manifolds, which are spaces that locally resemble Euclidean space, and introduces the idea of charts and atlases to describe manifolds. It also explains the tangent space as the space of tangent vectors to curves on a manifold, and discusses the change of coordinates and basis in vector spaces. Additionally, the lecture delves into tensors as linear maps and the metric, which defines the line element in a manifold. The instructor provides insights on how to define lengths and distances in a curved space.