Explore the mathematical foundations of Riemannian geometry through this 42-minute lecture that delves into advanced concepts essential for understanding curved spacetime in general relativity. Learn how the geodesic equation transitions to its Newtonian limit, providing crucial insight into the relationship between Einstein's theory and classical mechanics. Master the concept of affine connections, which describe how vectors change as they are transported along curves in curved manifolds. Develop proficiency in covariant differentiation, a fundamental tool that extends ordinary calculus to curved spaces by accounting for the geometry of the manifold. Gain the mathematical framework necessary for advanced studies in differential geometry, general relativity, and theoretical physics through rigorous treatment of these interconnected topics.