be able to account for the definition a differentiable manifold and be able to describe the most common families of Lie groups.

be able to account for the definition of the tangent space of a differentiable manifold, and give examples.

be able to explain the concepts immersion, embedding and submersion, and state different versions of the inverse mapping theorem.

be able to account for the definition of the tangent bundle of a differentiable manifold, with examples, and explain the relation between Lie algebras and Lie groups.

be able to account for the definitions of a Riemann metric and of a Riemannian manifold, and give some examples.

be able to account for the definition of the Levi-Civita connection on a Riemannian manifold, and explain why it is useful.

be able to briefly account for the theory of geodesics on Riemannian manifolds, and for polar coordinates.

be able to state some properties of the Riemann curvature tensor.