Explore the fundamental concepts of differential geometry in this graduate-level lecture from the Warsaw4PhD and GeoPlanet PhD schools. Delve into the intricacies of covariant derivatives, connection coefficients, and the Levi-Civita connection. Examine Christoffel symbols and their significance in curved spaces. Investigate locally flat coordinates and the properties of covariant derivatives. Understand parallel transport and its applications. Study geodesics and their variational principles. Apply these concepts to practical examples, including Christoffel symbols on a 2-sphere and in the Newtonian approximation. Gain a comprehensive understanding of these advanced mathematical tools essential for theoretical physics and cosmology.