Riemannian metrics and gradients: Computing gradients from extensions

This lecture covers the computation of gradients on Riemannian manifolds, focusing on computing gradients through extensions and retractions. It discusses the relationship between gradients on Riemannian submanifolds and Euclidean spaces, emphasizing the use of orthogonal projectors to tangent spaces. The lecture provides examples of open manifolds and spheres in Euclidean spaces, illustrating the smooth extension and projection properties. Key concepts include the pullback of gradients, smooth extensions, orthogonal projectors, and self-adjointness of projectors.