Topological, differentiable and riemannian manifolds. Differentiable maps. Tangent and cotangent bundles. Tensor algebra. Vector fields and their flows; Lie derivative. Differential forms; de Rham complex. Orientable manifolds. Riemannian metrics.Curves and surfaces in tridimentional Euclidean space.Gauss curvature and theorema egregium. Surfaces of constant curvature. Covariant derivative and Riemann curvature tensor; geodesics. Normal coordinates. Lie groups and their Lie algebras; exponenential maps.