Lecture notes for a term-long course on differentiable manifolds. The main reference is, as is usual, Smooth Manifolds by Lee, which contains everything and more for a first, second, and maybe even a third course on manifolds! I also used an incredible set of handwritten notes which are available online, from Kevin Wortman, at the University of Utah.
As for prerequisites, I believe that this course should (in theory) be accessible to any Masters student who has a basic undergraduate maths background.
Basic definitions, examples, and the concept of a manifold. Coordinate charts, atlases, and differentiable structures.
Maps, tangent vectors, smooth functions, and derivatives of smooth maps.
Tangent vectors, cotangent vectors, one-forms, and vector bundles.
Tangent bundles, cotangent bundles, and partitions of unity.
Submersions, immersions, embeddings, submanifolds, and Sard’s theorem.
Inverse & implicit function theorems, rank theorem, Whitney embedding, and regular value theorem.
Vector fields, Lie brackets, flows, Lie groups as manifolds, and the exponential map.
Integral curves, flows, and the pushforward of vector fields.
Exterior algebra, differential forms, wedge product, and Stokes’ theorem.
Tensors, exterior forms, wedge product, and pullbacks of exterior forms.
Exterior differentiation, Lie derivatives, and De Rham cohomology.
Cohomology, Poincaré duality, Mayer–Vietoris sequence, orientations, integration on manifolds.
Cobordism, boundary operator, and degree of a smooth map.
Riemannian metrics and Riemannian submanifolds.
Geodesics, curvature, Gauss–Bonnet theorem, isometries, and volume forms.