Tangent vectors without embedding space: Equivalence classes

This lecture explores tangent vectors without an embedding space, focusing on tangent vectors as equivalence classes. By defining an equivalence relation on smooth curves, the lecture illustrates how the tangent space at a point on a manifold can be constructed. The equivalence relation is shown to be independent of the choice of chart around the point, leading to the concept of a tangent vector as an equivalence class of curves. The lecture emphasizes the abstract nature of tangent vectors and their role in optimization on manifolds, providing insights that can be applied to general manifolds.