Connections: Axiomatic Definition

This lecture covers the concept of connections on manifolds, focusing on differentiating vector fields and the properties of derivatives. It introduces an axiomatic definition of connections as maps that ensure tangency between vector fields. The instructor explains how connections can be defined on a manifold and highlights the importance of valid connections. Various properties and rules related to differentiating vector fields are discussed, emphasizing the role of connections in this process. The lecture concludes by stating that every manifold has multiple valid connections, providing insights into the fundamental aspects of connections in optimization on manifolds.