Riemannian Hessians: Connections and Symmetry

This lecture introduces the concept of connections on a manifold, defined as maps that preserve tangency between smooth vector fields. The fundamental theorem of Riemannian geometry is presented, showing the existence of a unique symmetric connection compatible with the metric. The lecture also covers the action of vector fields on functions, Lie brackets, and the notion of torsion-free and symmetric connections. Finally, the compatibility of connections with the metric on a Riemannian manifold is discussed.