Differentiable curves. Differentiable surfaces. Metric structures on surfaces. Extrinsic properties of an embedded surface. Intrinsic properties of a Riemannian surface. The Gauss-Bonnet Theorem. Abstract manifolds. Tensor calculus. Differential forms. Differential operators on Riemannian manifolds. Additional topic chosen by the instructor such as: curvature of Riemannian manifolds; exterior calculus; applications. There will be laboratory sessions with examples and applications.