Differentiable manifolds and maps: Definition and examples, Inverse and implicit function theorem, Submanifolds, immersions and submersions.The tangent and cotangent bundle: Vector bundles, (co)tangent bundle as a vector bundle, Vector fields, flows, Lie derivative.Differential forms and Integration: Exterior differential, closed and exact forms, Poincare' lemma, Integration on manifolds, Stokes theorem, De Rham cohomology.
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- Louis Auslander and Robert E. MacKenzie, “Introduction to differentiable manifolds”, Dover, 2nd edition, 2009.