Curves on a surface, tangent vector, tangent space at a point. Vector fields, tangent vectors as directional derivatives. Lie bracket of vector fields. Decomposition into tangent and normal components. Covariant derivative and second fundamental form.
Covariant derivative is intrinsic.
Geodesics-local existence and uniqueness theorem. Curvature and shape operator. More examples Theorem egregium of Gauss. Poincare model of non-Euclidean geometry. Vector fields and Lie brackets.