Curves in 3-dimensional Euclidean space. The canonical representation of a curve. The fundamental theorem.
Surfaces in 3-dimensional Euclidean space. Tangent plane. The first fundamental form. Isometries. Normal and geodesic curvature. Weingarten map, Gaussian curvature, mean curvature, principal curvatures. Christoffel symbols, geodesic lines, the Gauss-Bonnet formula, Riemann curvature tensor. Theorema Egregium.
3. Riemannian manifolds
Generalities about differential manifolds. Examples: matrix groups and surfaces in R^3. Tangent spaces of a manifold. Riemannian manifolds. Parallel transport and linear connections. The Jacobi equation and the Riemann curvature tensor. Other areas of modern differential geometry.