Integral Curves and Exponential Map

This lecture covers the concept of integral curves, which are paths on a manifold that follow the direction of a vector field. The instructor explains how integral curves relate to the flow domain and the uniqueness of maximum integral curves. Additionally, the lecture delves into the exponential map in Lie groups, showcasing its importance in connecting Lie algebras and groups. The exponential map is demonstrated through the matrix exponential, emphasizing its properties when dealing with matrices that commute. The lecture concludes with a detailed proof of a proposition regarding the relationship between the exponential map and one-parameter subgroups.