Riemannian metrics and gradients: Why and definition of Riemannian manifolds

This lecture introduces the concept of Riemannian metrics and gradients, focusing on their importance and definition in the context of Riemannian manifolds. Starting with a reminder of Euclidean gradients in a linear space, the instructor explains the smoothness of functions and their gradients. The lecture then delves into the notion of vector fields on manifolds, discussing their smoothness and extension properties. Additionally, the importance of inner products on tangent spaces and the definition of metrics on manifolds are explored. Finally, the concept of Riemannian manifolds is introduced, emphasizing the smooth variation of metrics with respect to vector fields. Examples, definitions, and key properties are highlighted throughout the lecture.