Differentiable manifolds: local structure, submanifolds, Sard's Theorem, transversality, vector fields and flows. Fibre bundles, tangent and cotangent bundles of a manifold. Lie derivative of vector fields. Lie algebras. Lie groups (classical). Homogeneous spaces. Differential forms, exterior derivative. Symplectic forms. Integration on manifolds. Stokes' theorem. One or more of the following additional topics may also be covered:
- Riemannian manifolds. Curvature. Symmetric spaces. Classical examples.
- de Rham cohomology, singular cohomology and de Rham's theorem.
- Degree of a map. Index of a vector field. Applications.
- Distributions. Frobenius and Stefan-Sussmann theorems.