 Smooth Manifolds: Topological manifolds, smooth structures and local coordinates, examples.
 Smooth Maps: Smooth functions, partitions of unity.
 Tangent Bundle: Tangent vectors, tangent spaces, computation in local coordinates, tangent bundle, tangent of mappings, cotangent bundle; introduction to vector bundle.
 Submanifolds: Submersions, immersions and embeddings; submanifolds; inverse function theorem.
 Vector Fields: Vector fields and integral curves, flows, fundamental theorem on flows, complete vector fields; Lie derivative and Lie bracket, basic properties.
 Tensors: Algebra of tensors, tensors and tensor fields on manifolds, symmetric tensors, Riemannian metrics.
 Differential Forms: Exterior algebra, differential forms on manifolds, exterior derivatives; closed and exact forms, Poincare lemma; symplectic forms, Darboux theorem.
 Integration on Manifolds: Orientations, integration of differential forms, Stokes' theorem.
